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== 3.4#h(0.6em)Graph Linear Inequalities in Two Variables

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Graph #math.equation(block: false, alt: "x greater than 2")[$x > 2$] on a number line. #linebreak() 
If you missed this problem, review .

#solutionbox[
#figure(figph[A number line displays the inequality x \> 2. It shows an open circle at 2 and a thick arrow extending to the right, indicating all real numbers greater than 2. The number line ranges from -5 to 5.], alt: "A number line displays the inequality x > 2. It shows an open circle at 2 and a thick arrow extending to the right, indicating all real numbers greater than 2. The number line ranges from -5 to 5.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 x plus 3 greater than 23")[$4 x + 3 > 23$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x greater than 5")[$x > 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate: #math.equation(block: false, alt: "8 greater than x greater than 3")[$8 > x > 3$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: false, alt: "x")[$x$] is between 3 and 8, not inclusive
]
]


=== Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at #strong[linear inequalities] in two variables which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Linear Inequality]

A #strong[linear inequality] is an inequality that can be written in one of the following forms:

#math.equation(block: true, alt: "A x plus B y greater than C, A x plus B y greater than or equal to C, A x plus B y less than C, A x plus B y less than or equal to C")[$A x + B y > C &  &  & A x + B y ≥ C &  &  & A x + B y < C &  &  & A x + B y ≤ C$]Where #emph[A] and #emph[B] are not both zero.
]

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3.

#figure(figph[Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.], alt: "Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.", caption: none)

Similarly, linear inequalities in two variables have many solutions. Any ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that makes an inequality true when we substitute in the values is a #strong[solution to a linear inequality].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solution to a Linear Inequality]

An ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a #strong[solution to a linear inequality] if the inequality is true when we substitute the values of #emph[x] and #emph[y].
]

#examplebox("Example 1")[][
Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4 :")[$y > x + 4 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]], [#figure(figph[A mathematical inequality is displayed, showing 'y \> x + 4' in black text against a white background.], alt: "A mathematical inequality is displayed, showing 'y > x + 4' in black text against a white background.", caption: none)]),
  [#figure(figph[Text instruction: 'Substitute 0 for x and 0 for y.', with the first '0' in red and the second '0' in light blue.], alt: "Text instruction: 'Substitute 0 for x and 0 for y.', with the first '0' in red and the second '0' in light blue.", caption: none)

$#h(1em)$], [#figure(figph[The expression '0 \>? 0 + 4' questions whether zero is greater than four, with a question mark above the inequality symbol.], alt: "The expression '0 >? 0 + 4' questions whether zero is greater than four, with a question mark above the inequality symbol.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression '0 not greater than or equal to 4' is displayed on a white background.], alt: "The mathematical expression '0 not greater than or equal to 4' is displayed on a white background.", caption: none)],
  [], [So, #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$]], [#figure(figph[The image shows the mathematical inequality y \> x + 4.], alt: "The image shows the mathematical inequality y > x + 4.", caption: none)]),
  [#figure(figph[The text 'Substitute 1 for x and 6 for y' is displayed on a white background. The number 1 is highlighted in red, and the number 6 is highlighted in light blue.], alt: "The text 'Substitute 1 for x and 6 for y' is displayed on a white background. The number 1 is highlighted in red, and the number 6 is highlighted in light blue.", caption: none)

$#h(1em)$], [#figure(figph[An inequality 6 \> 1 + 4 with a question mark, prompting evaluation of whether six is indeed greater than the sum of one and four.], alt: "An inequality 6 > 1 + 4 with a question mark, prompting evaluation of whether six is indeed greater than the sum of one and four.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical inequality showing that 6 is greater than 5, represented as '6 \> 5' in a bold, dark gray font on a white background.], alt: "A simple mathematical inequality showing that 6 is greater than 5, represented as '6 > 5' in a bold, dark gray font on a white background.", caption: none)],
  [], [So, #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] is a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]], [#figure(figph[The mathematical inequality y \> x + 4 is displayed in black text on a white background, representing a region above the line y = x + 4 in a coordinate plane.], alt: "The mathematical inequality y > x + 4 is displayed in black text on a white background, representing a region above the line y = x + 4 in a coordinate plane.", caption: none)]),
  [#figure(figph[Substitute 2 for x and 6 for y.], alt: "Substitute 2 for x and 6 for y.", caption: none)

$#h(1em)$], [#figure(figph[A math problem featuring the inequality '6 \> 2 + 4' with a question mark over the greater-than sign, asking if the statement is true. The number 6 is blue and 2 is red.], alt: "A math problem featuring the inequality '6 > 2 + 4' with a question mark over the greater-than sign, asking if the statement is true. The number 6 is blue and 2 is red.", caption: none)],
  [Simplify.], [#figure(figph["6 is not greater than 6"], alt: "\"6 is not greater than 6\"", caption: none)],
  [], [So, #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]],
))

ⓓ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$]], [#figure(figph[The mathematical inequality 'y \> x + 4' is displayed on a white background.], alt: "The mathematical inequality 'y > x + 4' is displayed on a white background.", caption: none)]),
  [#figure(figph[Substitute -5 for x and -15 for y.], alt: "Substitute -5 for x and -15 for y.", caption: none)], [#figure(figph[A mathematical inequality problem asks to determine if -15 is greater than -5 + 4, with a question mark above the greater than symbol, implying verification of the statement.], alt: "A mathematical inequality problem asks to determine if -15 is greater than -5 + 4, with a question mark above the greater than symbol, implying verification of the statement.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the mathematical inequality -15 not greater than or equal to -1, which is equivalent to -15 \< -1.], alt: "The image displays the mathematical inequality -15 not greater than or equal to -1, which is equivalent to -15 < -1.", caption: none)],
  [], [So, #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] is not a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]],
))

ⓔ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$]], [#figure(figph[The mathematical inequality y \> x + 4 is shown in black text on a white background.], alt: "The mathematical inequality y > x + 4 is shown in black text on a white background.", caption: none)]),
  [#figure(figph[The image displays text instructions to 'Substitute -8 for x and 12 for y.', with -8 highlighted in red and 12 highlighted in light blue.], alt: "The image displays text instructions to 'Substitute -8 for x and 12 for y.', with -8 highlighted in red and 12 highlighted in light blue.", caption: none)

$#h(0.3em)$], [#figure(figph[An inequality expression showing '12 \> ? -8 + 4' on a white background. The number '12' is light blue, '-8' is red, and the question mark is above the greater than symbol.], alt: "An inequality expression showing '12 > ? -8 + 4' on a white background. The number '12' is light blue, '-8' is red, and the question mark is above the greater than symbol.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical inequality '12 \> -4' is displayed in a simple, clear font against a white background.], alt: "The mathematical inequality '12 > -4' is displayed in a simple, clear font against a white background.", caption: none)],
  [], [So, #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] is a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 3 :")[$y > x − 3 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 , 9 close parenthesis")[$( 4 , 9 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$]

#solutionbox[
ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x plus 1 :")[$y < x + 1 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 , 6 close parenthesis")[$( 8 , 6 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]

#solutionbox[
ⓐ yes ⓑ yes ⓒ no ⓓ no #linebreak() ⓔ yes
]
]


=== Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in shown previously again. The point #math.equation(block: false, alt: "x equals 3")[$x = 3$] separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3.

#figure(figph[Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled “numbers greater than 3”. The part of the number line to the left of 3 is labeled “numbers less than 3”.], alt: "Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled “numbers greater than 3”. The part of the number line to the left of 3 is labeled “numbers less than 3”.", caption: [The solution to #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is the shaded part of the number line to the right of #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$]])

Similarly, the line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] separates the plane into two regions. On one side of the line are points with #math.equation(block: false, alt: "y less than x plus 4 .")[$y < x + 4 .$] On the other side of the line are the points with #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] We call the line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] a #strong[boundary line].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Boundary Line]

The line with equation #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$] is the #strong[boundary line] that separates the region where #math.equation(block: false, alt: "A x plus B y greater than C")[$A x + B y > C$] from the region where #math.equation(block: false, alt: "A x plus B y less than C .")[$A x + B y < C .$]
]

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not #emph[a] is included in the solution:

#figure(figph[Two number lines are shown with the middle labeled with the number “a”. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled “x is less than a” and the number a is marked with an open parenthesis. The second number line is labeled “x is less than or equal to a” and the number a is marked with an open bracket.], alt: "Two number lines are shown with the middle labeled with the number “a”. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled “x is less than a” and the number a is marked with an open parenthesis. The second number line is labeled “x is less than or equal to a” and the number a is marked with an open bracket.", caption: none)

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

#math.equation(block: true, alt: "A x plus B y less than C, A x plus B y less than or equal to C; A x plus B y greater than C, A x plus B y greater than or equal to C; Boundary line is A x plus B y equals C, Boundary line is A x plus B y equals C; Boundary line is not included in solution., Boundary line is included in solution.; Boundary line is dashed., Boundary line is solid.")[$A x + B y < C &  &  &  &  & A x + B y ≤ C \ A x + B y > C &  &  &  &  & A x + B y ≥ C \ "Boundary line is" #h(0.2em) A x + B y = C &  &  &  &  & "Boundary line is" #h(0.2em) A x + B y = C \ "Boundary line is not included in solution." &  &  &  &  & "Boundary line is included in solution." \ "Boundary line is dashed." &  &  &  &  & "Boundary line is solid."$]Now, let’s take a look at what we found in . We’ll start by graphing the line #math.equation(block: false, alt: "y equals x plus 4 ,")[$y = x + 4 ,$] and then we’ll plot the five points we tested, as shown in the graph.

#figure(figph[This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).], alt: "This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).", caption: none)

In  we found that some of the points were solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$] and some were not.

Which of the points we plotted are solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4 ?")[$y > x + 4 ?$]

The points #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] and #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] are solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] Notice that they are both on the same side of the boundary line #math.equation(block: false, alt: "y equals x plus 4 .")[$y = x + 4 .$]

The two points #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis")[$( −5 , −15 )$] are on the other side of the boundary line #math.equation(block: false, alt: "y equals x plus 4 ,")[$y = x + 4 ,$] and they are not solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] For those two points, #math.equation(block: false, alt: "y less than x plus 4 .")[$y < x + 4 .$]

What about the point #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis ?")[$( 2 , 6 ) ?$] Because #math.equation(block: false, alt: "6 equals 2 plus 4 ,")[$6 = 2 + 4 ,$] the point is a solution to the equation #math.equation(block: false, alt: "y equals x plus 4 ,")[$y = x + 4 ,$] but not a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] So the point #math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] is on the boundary line.

Let’s take another point above the boundary line and test whether or not it is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] The point #math.equation(block: false, alt: "open parenthesis 0 , 10 close parenthesis")[$( 0 , 10 )$] clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?

#math.equation(block: true, alt: "y, greater than, x plus 4; 10, greater than ?, 0 plus 4; 10, greater than, 4")[$y & > & x + 4 \ 10 & limits(>)^(?) & 0 + 4 \ 10 & > & 4$]So, #math.equation(block: false, alt: "open parenthesis 0 , 10 close parenthesis")[$( 0 , 10 )$] is a solution to #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]

Any point you choose above the boundary line is a solution to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$] All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −15 close parenthesis ,")[$( −5 , −15 ) ,$] are not solutions to #math.equation(block: false, alt: "y greater than x plus 4 ,")[$y > x + 4 ,$] as shown.

#figure(figph[This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.], alt: "This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.", caption: none)

The graph of the inequality #math.equation(block: false, alt: "y greater than x plus 4")[$y > x + 4$] is shown in below.

The line #math.equation(block: false, alt: "y equals x plus 4")[$y = x + 4$] divides the plane into two regions. The shaded side shows the solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4 .")[$y > x + 4 .$]

The points on the boundary line, those where #math.equation(block: false, alt: "y equals x plus 4 ,")[$y = x + 4 ,$] are not solutions to the inequality #math.equation(block: false, alt: "y greater than x plus 4 ,")[$y > x + 4 ,$] so the line itself is not part of the solution. We show that by making the line dashed, not solid.

#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#examplebox("Example 2")[][
The boundary line shown in this graph is #math.equation(block: false, alt: "y equals 2 x minus 1 .")[$y = 2 x − 1 .$] Write the inequality shown by the graph.

#figure(figph[ This figure has the graph of a straight solid line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight solid line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight solid line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight solid line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
The line #math.equation(block: false, alt: "y equals 2 x minus 1")[$y = 2 x − 1$] is the boundary line. On one side of the line are the points with #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$] and on the other side of the line are the points with #math.equation(block: false, alt: "y less than 2 x minus 1 .")[$y < 2 x − 1 .$]

Let’s test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and see which inequality describes its position relative to the boundary line.

At #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] which inequality is true: #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$] or #math.equation(block: false, alt: "y less than 2 x minus 1 ?")[$y < 2 x − 1 ?$]

#math.equation(block: true, alt: "y greater than 2 x minus 1, y less than 2 x minus 1; 0 greater than ? 2 times 0 minus 1, 0 less than ? 2 times 0 minus 1; 0 greater than −1 True, 0 less than −1 False")[$y > 2 x − 1 &  &  &  &  & y < 2 x − 1 \ 0 limits(>)^(?) 2 · 0 − 1 &  &  &  &  & 0 limits(<)^(?) 2 · 0 − 1 \ 0 > −1 #h(0.2em) "True" &  &  &  &  & 0 < −1 #h(0.2em) "False"$]Since, #math.equation(block: false, alt: "y greater than 2 x minus 1")[$y > 2 x − 1$] is true, the side of the line with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] is the solution. The shaded region shows the solution of the inequality #math.equation(block: false, alt: "y greater than 2 x minus 1 .")[$y > 2 x − 1 .$]

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality #math.equation(block: false, alt: "y greater than or equal to 2 x minus 1 .")[$y ≥ 2 x − 1 .$]

We could use any point as a test point, provided it is not on the line. Why did we choose #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ?")[$( 0 , 0 ) ?$] Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that #math.equation(block: false, alt: "y less than 2 x minus 1 .")[$y < 2 x − 1 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals −2 x plus 3 .")[$y = −2 x + 3 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y greater than or equal to −2 x plus 3")[$y ≥ −2 x + 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 4 .")[$y = frac(1, 2) x − 4 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y less than or equal to the fraction 1 over 2 x minus 4")[$y ≤ frac(1, 2) x − 4$]
]
]

#examplebox("Example 3")[][
The boundary line shown in this graph is #math.equation(block: false, alt: "2 x plus 3 y equals 6 .")[$2 x + 3 y = 6 .$] Write the inequality shown by the graph.

#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
The line #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$] is the boundary line. On one side of the line are the points with #math.equation(block: false, alt: "2 x plus 3 y greater than 6")[$2 x + 3 y > 6$] and on the other side of the line are the points with #math.equation(block: false, alt: "2 x plus 3 y less than 6 .")[$2 x + 3 y < 6 .$]

Let’s test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and see which inequality describes its side of the boundary line.

At #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] which inequality is true: #math.equation(block: false, alt: "2 x plus 3 y greater than 6")[$2 x + 3 y > 6$] or #math.equation(block: false, alt: "2 x plus 3 y less than 6 ?")[$2 x + 3 y < 6 ?$]

#math.equation(block: true, alt: "2 x plus 3 y, greater than, 6; 2 open parenthesis 0 close parenthesis plus 3 open parenthesis 0 close parenthesis, greater than ?, 6; 0, greater than, 6 False, 2 x plus 3 y, less than, 6; 2 open parenthesis 0 close parenthesis plus 3 open parenthesis 0 close parenthesis, less than ?, 6; 0, less than, 6 True")[$2 x + 3 y & > & 6 \ 2 ( 0 ) + 3 ( 0 ) & limits(>)^(?) & 6 \ 0 & > & 6 #h(0.2em) "False" &  &  &  &  & 2 x + 3 y & < & 6 \ 2 ( 0 ) + 3 ( 0 ) & limits(<)^(?) & 6 \ 0 & < & 6 #h(0.2em) "True"$]So the side with #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is the side where #math.equation(block: false, alt: "2 x plus 3 y less than 6 .")[$2 x + 3 y < 6 .$]

(You may want to pick a point on the other side of the boundary line and check that #math.equation(block: false, alt: "2 x plus 3 y greater than 6 .")[$2 x + 3 y > 6 .$])

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The shaded region shows the solution to the inequality #math.equation(block: false, alt: "2 x plus 3 y less than 6 .")[$2 x + 3 y < 6 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x minus 4 y equals 8 .")[$x − 4 y = 8 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "x minus 4 y less than or equal to 8")[$x − 4 y ≤ 8$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "3 x minus y equals 6 .")[$3 x − y = 6 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "3 x minus y greater than or equal to 6")[$3 x − y ≥ 6$]
]
]


=== Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

#examplebox("Example 4")[How to Graph a Linear Equation in Two Variables][
Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to the fraction 3 over 4 x minus 2 .")[$y ≥ frac(3, 4) x − 2 .$]

#solutionbox[
#figure(figph[Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).], alt: "Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).", caption: none)

#figure(figph[Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.], alt: "Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.", caption: none)

#figure(figph[Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.], alt: "Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality #math.equation(block: false, alt: "y greater than or equal to the fraction 5 over 2 x minus 4 .")[$y ≥ frac(5, 2) x − 4 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (2, 1), and (4, 6). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region and on the boundary line, represent the solutions to #math.equation(block: false, alt: "y greater than the fraction 5 over 2 x minus 4 .")[$y > frac(5, 2) x − 4 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality #math.equation(block: false, alt: "y less than the fraction 2 over 3 x minus 5 .")[$y < frac(2, 3) x − 5 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 5), (3, negative 3), and (5, negative 1). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "y less than the fraction 2 over 3 x minus 5 .")[$y < frac(2, 3) x − 5 .$]
]
]

The steps we take to graph a linear inequality are summarized here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Graph a linear inequality in two variables.]

+ Identify and graph the boundary line.
- If the inequality is #math.equation(block: false, alt: "less than or equal to or greater than or equal to ,")[$≤ "or" ≥ ,$] the boundary line is solid.
- If the inequality is #math.equation(block: false, alt: "less than or greater than ,")[$< "or" > ,$] the boundary line is dashed.
+ Test a point that is not on the boundary line. Is it a solution of the inequality?
+ Shade in one side of the boundary line.
- If the test point is a solution, shade in the side that includes the point.
- If the test point is not a solution, shade in the opposite side.
]

#examplebox("Example 5")[][
Graph the linear inequality #math.equation(block: false, alt: "x minus 2 y less than 5 .")[$x − 2 y < 5 .$]

#solutionbox[
First, we graph the boundary line #math.equation(block: false, alt: "x minus 2 y equals 5 .")[$x − 2 y = 5 .$] The inequality is #math.equation(block: false, alt: "less than")[$<$] so we draw a dashed line.

#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).", caption: none)

Then, we test a point. We’ll use #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] again because it is easy to evaluate and it is not on the boundary line.

Is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] a solution of #math.equation(block: false, alt: "x minus 2 y less than 5 ?")[$x − 2 y < 5 ?$]

#figure(figph[Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.], alt: "Is 0 minus 2 times 0 less than 5? Is 0 minus 0 less than 5? 0 is less than 5.", caption: none)

The point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is a solution of #math.equation(block: false, alt: "x minus 2 y less than 5 ,")[$x − 2 y < 5 ,$] so we shade in that side of the boundary line.

#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "x minus 2 y less than 5 .")[$x − 2 y < 5 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "2 x minus 3 y less than 6 .")[$2 x − 3 y < 6 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 2), (3, 0), and (6, 2). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "2 x minus 3 y less than 6 .")[$2 x − 3 y < 6 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "2 x minus y greater than 3 .")[$2 x − y > 3 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 1), and (2, 1). The line divides the x y-coordinate plane into two halves. The bottom right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "2 x minus y greater than 3 .")[$2 x − y > 3 .$]
]
]

What if the boundary line goes through the origin? Then, we won’t be able to use #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

#examplebox("Example 6")[][
Graph the linear inequality: #math.equation(block: false, alt: "y less than or equal to ​ minus 4 x .")[$y ≤ "​" − 4 x .$]

#solutionbox[
First, we graph the boundary line #math.equation(block: false, alt: "y equals −4 x .")[$y = −4 x .$] It is in slope–intercept form, with #math.equation(block: false, alt: "m equals −4")[$m = −4$] and #math.equation(block: false, alt: "b equals 0 .")[$b = 0 .$] The inequality is #math.equation(block: false, alt: "less than or equal to")[$≤$] so we draw a solid line.

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).", caption: none)

Now we need a test point. We can see that the point #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] is not on the boundary line.

Is #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] a solution of #math.equation(block: false, alt: "y less than or equal to −4 x ?")[$y ≤ −4 x ?$]

#figure(figph[Mathematical expressions demonstrating 0 is less than or equal to -4(1) with a red question mark, which simplifies to the false statement 0 \<= -4, followed by the true statement 0 != -4.], alt: "Mathematical expressions demonstrating 0 is less than or equal to -4(1) with a red question mark, which simplifies to the false statement 0 <= -4, followed by the true statement 0 != -4.", caption: none)

The point #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$] is not a solution to #math.equation(block: false, alt: "y less than or equal to ​ minus 4 x ,")[$y ≤ "​" − 4 x ,$] so we shade in the opposite side of the boundary line.

#figure(figph[Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.], alt: "Is 0 less than or equal to negative 4 times 1? 0 is not less than or equal to negative 4.", caption: none)

All points in the shaded region and on the boundary line represent the solutions to #math.equation(block: false, alt: "y less than or equal to ​ minus 4 x .")[$y ≤ "​" − 4 x .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "y greater than minus 3 x .")[$y > − 3 x .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 3), (0, 0), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "y greater than minus 3 x .")[$y > − 3 x .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "y greater than or equal to −2 x .")[$y ≥ −2 x .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (negative 1, 2), (0, 0), and (1, negative 2). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region and on the boundary line, represent the solutions to #math.equation(block: false, alt: "y greater than or equal to −2 x .")[$y ≥ −2 x .$]
]
]

Some linear inequalities have only one variable. They may have an #emph[x] but no #emph[y], or a #emph[y] but no #emph[x]. In these cases, the boundary line will be either a vertical or a horizontal line.

Recall that:

#math.equation(block: true, alt: "x equals a, vertical line; y equals b, horizontal line")[$x = a &  &  &  &  & "vertical line" \ y = b &  &  &  &  & "horizontal line"$]#examplebox("Example 7")[][
Graph the linear inequality: #math.equation(block: false, alt: "y greater than 3 .")[$y > 3 .$]

#solutionbox[
First, we graph the boundary line #math.equation(block: false, alt: "y equals 3 .")[$y = 3 .$] It is a horizontal line. The inequality is #math.equation(block: false, alt: "greater than")[$>$] so we draw a dashed line.

We test the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$]

#math.equation(block: true, alt: "y greater than 3; 0 greater than 3")[$y > 3 \ 0 cancel(>) 3$]So, #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is not a solution to #math.equation(block: false, alt: "y greater than 3 .")[$y > 3 .$]

So we shade the side that does not include #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] as shown in this graph.

#figure(figph[This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "y greater than 3 .")[$y > 3 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "y less than 5 .")[$y < 5 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal dashed line is drawn through the points (negative 1, 5), (0, 5), and (1, 5). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
All points in the shaded region, but not those on the boundary line, represent the solutions to #math.equation(block: false, alt: "y less than 5 .")[$y < 5 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Graph the linear inequality: #math.equation(block: false, alt: "y less than or equal to −1 .")[$y ≤ −1 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A horizontal line is drawn through the points (negative 1, negative 1), (0, negative 1), and (1, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)


 #linebreak() 
All points in the shaded region and on the boundary line represent the solutions to #math.equation(block: false, alt: "y less than or equal to −1 .")[$y ≤ −1 .$]
]
]


=== Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

#examplebox("Example 8")[][
Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least \$240 a week. Her job in food service pays \$10 an hour and her tutoring job on campus pays \$15 an hour. How many hours does Hilaria need to work at each job to earn at least \$240?

ⓐ Let #math.equation(block: false, alt: "x")[$x$] be the number of hours she works at the job in food service and let #emph[y] be the number of hours she works tutoring. Write an inequality that would model this situation.

ⓑ Graph the inequality.

ⓒ Find three ordered pairs #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that would be solutions to the inequality. Then, explain what that means for Hilaria.

#solutionbox[
ⓐ We let #emph[x] be the number of hours she works at the job in food service and let #emph[y] be the number of hours she works tutoring.

She earns \$10 per hour at the job in food service and \$15 an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.

#figure(figph[10 x plus 15 y is greater than 240. The “10 x” is labeled “Amount earned at the food service job”. The “15 y” is labeled “the amount earned tutoring”. The “is greater than 240” is labeled “is at least 240”.], alt: "10 x plus 15 y is greater than 240. The “10 x” is labeled “Amount earned at the food service job”. The “15 y” is labeled “the amount earned tutoring”. The “is greater than 240” is labeled “is at least 240”.", caption: none)

ⓑ To graph the inequality, we put it in slope–intercept form.

#math.equation(block: true, alt: "10 x plus 15 y, greater than or equal to, 240; 15 y, greater than or equal to, −10 x plus 240; y, greater than or equal to, minus the fraction 2 over 3 x plus 16")[$10 x + 15 y & ≥ & 240 \ 15 y & ≥ & −10 x + 240 \ y & ≥ & − frac(2, 3) x + 16$]#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

ⓒ From the graph, we see that the ordered pairs #math.equation(block: false, alt: "open parenthesis 15 , 10 close parenthesis , open parenthesis 0 , 16 close parenthesis , open parenthesis 24 , 0 close parenthesis")[$( 15 , 10 ) , ( 0 , 16 ) , ( 24 , 0 )$] represent three of infinitely many solutions. Check the values in the inequality.

#figure(figph[First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.], alt: "First we test the point (15, 10) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 15 plus 15 times 10 greater than or equal to 240? Since 300 is greater than or equal to 240 (15, 10) is a solution. Next we test the point (0, 16) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 0 plus 15 times 16 greater than or equal to 240? Since 240 is greater than or equal to 240 (0, 16) is a solution. Then we test the point (24, 0) in the inequality 10 x plus 15 y greater than or equal to 240. Is 10 times 24 plus 15 times 0 greater than or equal to 240? Since 240 is greater than or equal to 240 (24, 0) is a solution.", caption: none)

For Hilaria, it means that to earn at least \$240, she can work 15 hours tutoring and 10 hours at her fast-food job, earn all her money tutoring for 16 hours, or earn all her money while working 24 hours at the job in food service.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Hugh works two part time jobs. One at a grocery store that pays \$10 an hour and the other is babysitting for \$13 hour. Between the two jobs, Hugh wants to earn at least \$260 a week. How many hours does Hugh need to work at each job to earn at least \$260?

ⓐ Let #emph[x] be the number of hours he works at the grocery store and let #emph[y] be the number of hours he works babysitting. Write an inequality that would model this situation.

ⓑ Graph the inequality.

ⓒ Find three ordered pairs (#emph[x], #emph[y]) that would be solutions to the inequality. Then, explain what that means for Hugh.

#solutionbox[
ⓐ #math.equation(block: false, alt: "10 x plus 13 y greater than or equal to 260")[$10 x + 13 y ≥ 260$] #linebreak() ⓑ  #linebreak() 
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 30. A line is drawn through the points (0, 20), (13, 10), and (26, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
ⓒ Answers will vary.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Veronica works two part time jobs in order to earn enough money to meet her obligations of at least \$280 a week. Her job at the day spa pays \$10 an hour and her administrative assistant job on campus pays \$17.50 an hour. How many hours does Veronica need to work at each job to earn at least \$280?

ⓐ Let #emph[x] be the number of hours she works at the day spa and let #emph[y] be the number of hours she works as administrative assistant. Write an inequality that would model this situation.

ⓑ Graph the inequality.

ⓒ Find three ordered pairs (#emph[x], #emph[y]) that would be solutions to the inequality. Then, explain what that means for Veronica

#solutionbox[
ⓐ #math.equation(block: false, alt: "10 x plus 17.5 y greater than or equal to 280")[$10 x + 17.5 y ≥ 280$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16) and (28, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
ⓒ Answers will vary.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with graphing linear inequalities in two variables.

- #link("https://openstax.org/l/37linineqgraphs")[Graphing Linear Inequalities in Two Variables]
]


=== Key Concepts

- #strong[How to graph a linear inequality in two variables.]
+ Identify and graph the boundary line. #linebreak() 
If the inequality is #math.equation(block: false, alt: "less than or equal to or greater than or equal to ,")[$≤ "or" ≥ ,$] the boundary line is solid. #linebreak() 
If the inequality is #math.equation(block: false, alt: "less than or greater than ,")[$< #h(0.2em) "or" > ,$] the boundary line is dashed.
+ Test a point that is not on the boundary line. Is it a solution of the inequality?
+ Shade in one side of the boundary line. #linebreak() 
If the test point is a solution, shade in the side that includes the point. #linebreak() 
If the test point is not a solution, shade in the opposite side.


==== Practice Makes Perfect

#strong[Verify Solutions to an Inequality in Two Variables]

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 1 :")[$y > x − 1 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$( −4 , −1 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]

#solutionbox[
ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ no
]

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y greater than x minus 3 :")[$y > x − 3 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis −6 , −3 close parenthesis")[$( −6 , −3 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$]

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than 3 x plus 2 :")[$y < 3 x + 2 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −2 close parenthesis")[$( −3 , −2 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis")[$( −1 , 4 )$]

#solutionbox[
ⓐ no ⓑ no ⓒ no ⓓ yes ⓔ no
]

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than minus 2 x plus 5 :")[$y < − 2 x + 5 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −6 , −2 close parenthesis")[$( −6 , −2 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis")[$( 5 , −4 )$]

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "3 x minus 4 y greater than 4 :")[$3 x − 4 y > 4 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]

#solutionbox[
ⓐ yes ⓑ no ⓒ no ⓓ yes ⓔ no
]

Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "2 x plus 3 y greater than 2 :")[$2 x + 3 y > 2 :$]

ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 4 , −3 close parenthesis")[$( 4 , −3 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis −8 , 12 close parenthesis")[$( −8 , 12 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]

#strong[Recognize the Relation Between the Solutions of an Inequality and its Graph]

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals 3 x minus 4 .")[$y = 3 x − 4 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y less than or equal to 3 x minus 4")[$y ≤ 3 x − 4$]
]

Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals 2 x minus 4 .")[$y = 2 x − 4 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x plus 1 .")[$y = frac(1, 2) x + 1 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 1), (2, 0), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y less than or equal to the fraction 1 over 2 x plus 1")[$y ≤ frac(1, 2) x + 1$]
]

Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals minus the fraction 1 over 3 x minus 2 .")[$y = − frac(1, 3) x − 2 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (3, negative 3), and (6, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals 5 .")[$x + y = 5 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 5), (1, 4), and (5, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "x plus y greater than or equal to 5")[$x + y ≥ 5$]
]

Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals 3 .")[$x + y = 3 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "3 x minus y equals 6 .")[$3 x − y = 6 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "3 x minus y less than or equal to 6")[$3 x − y ≤ 6$]
]

Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "2 x minus y equals 4 .")[$2 x − y = 4 .$]

#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 2), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

#strong[Graph Linear Inequalities in Two Variables]

In the following exercises, graph each linear inequality.

Graph the linear inequality: #math.equation(block: false, alt: "y greater than the fraction 2 over 3 x minus 1 .")[$y > frac(2, 3) x − 1 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y less than the fraction 3 over 5 x plus 2 .")[$y < frac(3, 5) x + 2 .$]

Graph the linear inequality: #math.equation(block: false, alt: "y less than or equal to minus the fraction 1 over 2 x plus 4 .")[$y ≤ − frac(1, 2) x + 4 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y greater than or equal to minus the fraction 1 over 3 x minus 2 .")[$y ≥ − frac(1, 3) x − 2 .$]

Graph the linear inequality: #math.equation(block: false, alt: "x minus y less than or equal to 3 .")[$x − y ≤ 3 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "x minus y greater than or equal to −2 .")[$x − y ≥ −2 .$]

Graph the linear inequality: #math.equation(block: false, alt: "4 x plus y greater than minus 4 .")[$4 x + y > − 4 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "x plus 5 y less than minus 5 .")[$x + 5 y < − 5 .$]

Graph the linear inequality: #math.equation(block: false, alt: "3 x plus 2 y greater than or equal to −6 .")[$3 x + 2 y ≥ −6 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "4 x plus 2 y greater than or equal to −8 .")[$4 x + 2 y ≥ −8 .$]

Graph the linear inequality: #math.equation(block: false, alt: "y greater than 4 x .")[$y > 4 x .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y less than or equal to −3 x .")[$y ≤ −3 x .$]

Graph the linear inequality: #math.equation(block: false, alt: "y less than minus 10 .")[$y < − 10 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, 3), and (1, negative 3). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y greater than or equal to 2 .")[$y ≥ 2 .$]

Graph the linear inequality: #math.equation(block: false, alt: "x less than or equal to 5 .")[$x ≤ 5 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight vertical dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A vertical dashed line is drawn through the points (5, negative 1), (5, 0), and (5, 1). The line divides the x y-coordinate plane into two halves. The left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "x greater than or equal to 0 .")[$x ≥ 0 .$]

Graph the linear inequality: #math.equation(block: false, alt: "x minus y less than 4 .")[$x − y < 4 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "x minus y less than minus 3 .")[$x − y < − 3 .$]

Graph the linear inequality: #math.equation(block: false, alt: "y greater than or equal to the fraction 3 over 2 x .")[$y ≥ frac(3, 2) x .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y less than or equal to the fraction 5 over 4 x .")[$y ≤ frac(5, 4) x .$]

Graph the linear inequality: #math.equation(block: false, alt: "y greater than minus 2 x plus 1 .")[$y > − 2 x + 1 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 1), (1, negative 1), and (2, negative 3). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "y less than minus 3 x minus 4 .")[$y < − 3 x − 4 .$]

Graph the linear inequality: #math.equation(block: false, alt: "2 x plus y greater than or equal to −4 .")[$2 x + y ≥ −4 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "x plus 2 y less than or equal to −2 .")[$x + 2 y ≤ −2 .$]

Graph the linear inequality: #math.equation(block: false, alt: "2 x minus 5 y greater than 10 .")[$2 x − 5 y > 10 .$]

#solutionbox[
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)
]

Graph the linear inequality: #math.equation(block: false, alt: "4 x minus 3 y greater than 12 .")[$4 x − 3 y > 12 .$]

#strong[Solve Applications using Linear Inequalities in Two Variables]

Harrison works two part time jobs. One at a gas station that pays \$11 an hour and the other is IT troubleshooting for #math.equation(block: false, alt: "$ 16.50")[$"$" 16.50$] an hour. Between the two jobs, Harrison wants to earn at least \$330 a week. How many hours does Harrison need to work at each job to earn at least \$330?

ⓐ Let #emph[x] be the number of hours he works at the gas station and let #emph[y] be the number of (hours he works troubleshooting. Write an inequality that would model this situation.

ⓑ Graph the inequality.

ⓒ Find three ordered pairs #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that would be solutions to the inequality. Then, explain what that means for Harrison.

#solutionbox[
ⓐ #math.equation(block: false, alt: "11 x plus 16.5 y greater than or equal to 330")[$11 x + 16.5 y ≥ 330$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() 
ⓒ Answers will vary.
]

Elena needs to earn at least \$450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays \$9 an hour and the other as an intern in a genetics lab for \$22.50 per hour. How many hours does Elena need to work at each job to earn at least \$450 per week?

ⓐ Let #emph[x] be the number of hours she works teaching swimming and let #emph[y] be the number of hours she works as an intern. Write an inequality that would model this situation.

ⓑ Graph the inequality.

ⓒ Find three ordered pairs #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that would be solutions to the inequality. Then, explain what that means for Elena.

The doctor tells Laura she needs to exercise enough to burn 500 calories each day. She prefers to either run or bike and burns 15 calories per minute while running and 10 calories a minute while biking.

ⓐ If #emph[x] is the number of minutes that Laura runs and #emph[y] is the number minutes she bikes, find the inequality that models the situation.

ⓑ Graph the inequality.

ⓒ List three solutions to the inequality. What options do the solutions provide Laura?

#solutionbox[
ⓐ #math.equation(block: false, alt: "15 x plus 10 y greater than or equal to 500")[$15 x + 10 y ≥ 500$] #linebreak() 
ⓑ  #linebreak() 
#figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 70. A line is drawn through the points (0, 50) and (34, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 70. A line is drawn through the points (0, 50) and (34, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none)

 #linebreak() ⓒ Answers will vary.
]

Armando’s workouts consist of kickboxing and swimming. While kickboxing, he burns 10 calories per minute and he burns 7 calories a minute while swimming. He wants to burn 600 calories each day.

ⓐ If #emph[x] is the number of minutes that Armando will kickbox and #emph[y] is the number minutes he will swim, find the inequality that will help Armando create a workout for today.

ⓑ Graph the inequality.

ⓒ List three solutions to the inequality. What options do the solutions provide Armando?


==== Writing Exercises

Lester thinks that the solution of any inequality with a #math.equation(block: false, alt: "greater than")[$>$] sign is the region above the line and the solution of any inequality with a #math.equation(block: false, alt: "less than")[$<$] sign is the region below the line. Is Lester correct? Explain why or why not.

#solutionbox[
Answers will vary.
]

Explain why, in some graphs of linear inequalities, the boundary line is solid but in other graphs it is dashed.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “verify solutions to an inequality in two variables.”, “recognize the relation between the solutions of an inequality and its graph”, and “graph linear inequalities”. The other columns are left blank so that the learner may indicate their mastery level for each topic.], alt: "This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “verify solutions to an inequality in two variables.”, “recognize the relation between the solutions of an inequality and its graph”, and “graph linear inequalities”. The other columns are left blank so that the learner may indicate their mastery level for each topic.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?