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== 3.5#h(0.6em)Solve Equations Using Integers; The Division Property of Equality

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

#math.equation(block: false, alt: "Evaluate x plus 4 when x equals −4 .")[$"Evaluate" #h(0.2em) x + 4 #h(0.2em) "when" #h(0.2em) x = −4 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "0")[$0$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Solve: y minus 6 equals 10 .")[$"Solve:" #h(0.2em) y − 6 = 10 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "16")[$16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic expression #math.equation(block: false, alt: "5")[$5$] #emph[less than] #math.equation(block: false, alt: "x .")[$x .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x minus 5")[$x − 5$]
]
]


=== Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[How to determine whether a number is a solution to an equation.]

+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
]

#examplebox("Example 1")[][
Determine whether each of the following is a solution of #math.equation(block: false, alt: "2 x minus 5 equals −13 :")[$2 x − 5 = −13 ":"$] #linebreak()

+ ⓐ #math.equation(block: false, alt: "x equals 4")[$#h(0.2em) x = 4 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −4")[$#h(0.2em) x = −4 #h(0.6em)$]
+ ⓒ #math.equation(block: false, alt: "x equals −9 .")[$#h(0.2em) x = −9 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ Substitute 4 for x in the equation to determine if it is true.], []),
  [], [#figure(figph[A mathematical equation is displayed: 2x - 5 = -13.], alt: "A mathematical equation is displayed: 2x - 5 = -13.", caption: none)],
  [#figure(figph[The text 'Substitute 4 for x.' is displayed in a dark teal-like font, with the number 4 highlighted in red.], alt: "The text 'Substitute 4 for x.' is displayed in a dark teal-like font, with the number 4 highlighted in red.", caption: none)], [#figure(figph[A mathematical equation displays '2(4) - 5 = -13,' with the number 4 highlighted in red, indicating a specific value in the problem.], alt: "A mathematical equation displays '2(4) - 5 = -13,' with the number 4 highlighted in red, indicating a specific value in the problem.", caption: none)],
  [Multiply.], [#figure(figph[The image presents the mathematical expression '8 - 5 ? = -13', with a question mark placed above the equals sign, indicating an inquiry into the truth of the statement.], alt: "The image presents the mathematical expression '8 - 5 ? = -13', with a question mark placed above the equals sign, indicating an inquiry into the truth of the statement.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical expression displays '3  eq -13', indicating that 3 is not equal to -13.], alt: "A mathematical expression displays '3 eq -13', indicating that 3 is not equal to -13.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals 4")[$x = 4$] does not result in a true equation, #math.equation(block: false, alt: "4")[$4$] is not a solution to the equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ Substitute −4 for x in the equation to determine if it is true.], [#figure(figph[A mathematical equation, 2x - 5 = -13, is displayed in a dark gray font against a plain white background.], alt: "A mathematical equation, 2x - 5 = -13, is displayed in a dark gray font against a plain white background.", caption: none)]),
  [#figure(figph[The text 'Substitute -4 for x.' is displayed in the image. The word 'Substitute' is in dark blue-gray, '-4' is in red, and 'for x.' is in dark blue-gray. The background is white.], alt: "The text 'Substitute -4 for x.' is displayed in the image. The word 'Substitute' is in dark blue-gray, '-4' is in red, and 'for x.' is in dark blue-gray. The background is white.", caption: none)], [#figure(figph[The image displays the mathematical expression 2(-4) - 5 ?= -13, where the question mark asks to verify the equality. Calculating the left side, 2 multiplied by -4 is -8, and then subtracting 5 yields -13. Thus, the equality holds true.], alt: "The image displays the mathematical expression 2(-4) - 5 ?= -13, where the question mark asks to verify the equality. Calculating the left side, 2 multiplied by -4 is -8, and then subtracting 5 yields -13. Thus, the equality holds true.", caption: none)],
  [Multiply.], [#figure(figph[The image displays a math problem questioning if -8 minus 5 is equal to -13. The calculation confirms that -8 - 5 does indeed equal -13.], alt: "The image displays a math problem questioning if -8 minus 5 is equal to -13. The calculation confirms that -8 - 5 does indeed equal -13.", caption: none)],
  [Subtract.], [#figure(figph[The mathematical equation -13 = -13 with a checkmark, indicating it is correct.], alt: "The mathematical equation -13 = -13 with a checkmark, indicating it is correct.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals −4")[$x = −4$] results in a true equation, #math.equation(block: false, alt: "−4")[$−4$] is a solution to the equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ Substitute −9 for x in the equation to determine if it is true.], []),
  [], [#figure(figph[The image shows a mathematical equation, 2x - 5 = -13, in black text on a white background.], alt: "The image shows a mathematical equation, 2x - 5 = -13, in black text on a white background.", caption: none)],
  [Substitute −9 for x.], [#figure(figph[A math equation \`2(-9) - 5 ?= -13\` is shown. The question mark in the equals sign asks to verify the equality. The number -9 is highlighted in red, drawing attention to its negative value.], alt: "A math equation `2(-9) - 5 ?= -13` is shown. The question mark in the equals sign asks to verify the equality. The number -9 is highlighted in red, drawing attention to its negative value.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression is displayed, '-18 - 5 ?= -13,' where the question mark above the equality sign indicates a verification of whether the left side is equal to the right side. -18 minus 5 equals -23.], alt: "A mathematical expression is displayed, '-18 - 5 ?= -13,' where the question mark above the equality sign indicates a verification of whether the left side is equal to the right side. -18 minus 5 equals -23.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical inequality states that -23 is not equal to -13, displayed in black text on a white background.], alt: "A mathematical inequality states that -23 is not equal to -13, displayed in black text on a white background.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals −9")[$x = −9$] does not result in a true equation, #math.equation(block: false, alt: "−9")[$−9$] is not a solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each of the following is a solution of #math.equation(block: false, alt: "2 x minus 8 equals −14 :")[$2 x − 8 = −14 ":"$]

+ ⓐ #math.equation(block: false, alt: "x equals −11")[$#h(0.2em) x = −11 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "x equals 11")[$#h(0.2em) x = 11 #h(0.6em)$]
+ ⓒ #math.equation(block: false, alt: "x equals −3")[$#h(0.2em) x = −3$]

#solutionbox[
+ ⓐ no
+ ⓑ no
+ ⓒ yes
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each of the following is a solution of #math.equation(block: false, alt: "2 y plus 3 equals −11 :")[$2 y + 3 = −11 ":"$]

+ ⓐ #math.equation(block: false, alt: "y equals 4")[$#h(0.2em) y = 4 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "y equals −4")[$#h(0.2em) y = −4 #h(0.6em)$]
+ ⓒ #math.equation(block: false, alt: "y equals −7")[$#h(0.2em) y = −7$]

#solutionbox[
+ ⓐ no
+ ⓑ no
+ ⓒ yes
]
]


=== Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

#figure(figph[This figure has two columns. The first column has the equation x plus 4 equals 12. Underneath there is x plus 4 minus 4 equals 12 minus 4. Under this there is x equals 8. The second column has the equation y minus 5 equals 9. Underneath there is the equation y minus 5 plus 5 equals 9 plus 5. Under this there is y equals 14.], alt: "This figure has two columns. The first column has the equation x plus 4 equals 12. Underneath there is x plus 4 minus 4 equals 12 minus 4. Under this there is x equals 8. The second column has the equation y minus 5 equals 9. Underneath there is the equation y minus 5 plus 5 equals 9 plus 5. Under this there is y equals 14.", caption: none)

When you add or subtract the same quantity from both sides of an equation, you still have equality.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Equalities]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Subtraction Property of Equality], [Addition Property of Equality]),
  [#math.equation(block: false, alt: "For any numbers a , b , c ,")[$"For any numbers" #h(0.2em) a , b , c ,$] #linebreak() #math.equation(block: false, alt: "if a equals b then a minus c equals b minus c .")[$"if" #h(0.2em) a = b #h(0.2em) "then" #h(0.2em) a − c = b − c .$]], [#math.equation(block: false, alt: "For any numbers a , b , c ,")[$"For any numbers" #h(0.2em) a , b , c ,$] #linebreak() #math.equation(block: false, alt: "if a equals b then a plus c equals b plus c .")[$"if" #h(0.2em) a = b #h(0.2em) "then" #h(0.2em) a + c = b + c .$]],
))
]

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "y plus 9 equals 5 .")[$y + 9 = 5 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing 'y + 9 = 5' in a clear, black font against a white background.], alt: "A mathematical equation is displayed, showing 'y + 9 = 5' in a clear, black font against a white background.", caption: none)]),
  [Subtract 9 from each side to undo the addition.], [#figure(figph[The mathematical equation y + 9 - 9 = 5 - 9 is displayed, demonstrating a step-by-step approach to isolate the variable y.], alt: "The mathematical equation y + 9 - 9 = 5 - 9 is displayed, demonstrating a step-by-step approach to isolate the variable y.", caption: none)],
  [Simplify.], [#figure(figph[The equation y = -4 is displayed in black text on a white background.], alt: "The equation y = -4 is displayed in black text on a white background.", caption: none)],
))

Check the result by substituting #math.equation(block: false, alt: "−4")[$−4$] into the original equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y plus 9 equals 5")[$y + 9 = 5 #h(1.4em)$]]),
  [Substitute −4 for y], [#math.equation(block: false, alt: "−4 plus 9 equals ? 5")[$−4 + 9 limits(=)^(?) 5 #h(1.4em)$]],
  [], [#math.equation(block: false, alt: "5 equals 5 ✓")[$5 = 5 ✓$]],
))

Since #math.equation(block: false, alt: "y equals −4")[$y = −4$] makes #math.equation(block: false, alt: "y plus 9 equals 5")[$y + 9 = 5$] a true statement, we found the solution to this equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "y plus 11 equals 7")[$y + 11 = 7$]

#solutionbox[
−4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "y plus 15 equals −4")[$y + 15 = −4$]

#solutionbox[
−19
]
]

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "a minus 6 equals −8")[$a − 6 = −8$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation displays 'a - 6 = -8' in black font on a white background. This is a linear equation with one variable 'a', where 6 is subtracted from 'a', resulting in -8.], alt: "A mathematical equation displays 'a - 6 = -8' in black font on a white background. This is a linear equation with one variable 'a', where 6 is subtracted from 'a', resulting in -8.", caption: none)]),
  [Add 6 to each side to undo the subtraction.], [#figure(figph[A mathematical equation showing 'a - 6 + 6 = -8 + 6', where the '+ 6' on both sides is highlighted in red, illustrating the addition property of equality to solve for 'a'.], alt: "A mathematical equation showing 'a - 6 + 6 = -8 + 6', where the '+ 6' on both sides is highlighted in red, illustrating the addition property of equality to solve for 'a'.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'a = -2' is displayed in a bold, black font on a plain white background, presenting a simple algebraic assignment of the value negative two to the variable 'a'.], alt: "The mathematical equation 'a = -2' is displayed in a bold, black font on a plain white background, presenting a simple algebraic assignment of the value negative two to the variable 'a'.", caption: none)],
  [Check the result by substituting #math.equation(block: false, alt: "−2")[$−2$] into the original equation:], [#figure(figph[The image shows the equation 'a - 6 = -8' in black text on a white background.], alt: "The image shows the equation 'a - 6 = -8' in black text on a white background.", caption: none)],
  [Substitute #math.equation(block: false, alt: "−2")[$−2$] for #math.equation(block: false, alt: "a")[$a$]], [#figure(figph[The mathematical expression '-2 - 6 = -8' is presented, with a question mark over the equal sign, indicating a query about its validity or prompting verification of the sum.], alt: "The mathematical expression '-2 - 6 = -8' is presented, with a question mark over the equal sign, indicating a query about its validity or prompting verification of the sum.", caption: none)],
  [], [#figure(figph[A mathematical equation shows '-8 = -8' with a black checkmark symbol, indicating the equality is correct. The numbers and symbols are rendered in a slightly bold, shadowed style against a white background.], alt: "A mathematical equation shows '-8 = -8' with a black checkmark symbol, indicating the equality is correct. The numbers and symbols are rendered in a slightly bold, shadowed style against a white background.", caption: none)],
))

The solution to #math.equation(block: false, alt: "a minus 6 equals −8")[$a − 6 = −8$] is #math.equation(block: false, alt: "−2 .")[$−2 .$]

Since #math.equation(block: false, alt: "a equals −2")[$a = −2$] makes #math.equation(block: false, alt: "a minus 6 equals −8")[$a − 6 = −8$] a true statement, we found the solution to this equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "a minus 2 equals −8")[$a − 2 = −8$]

#solutionbox[
−6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "n minus 4 equals −8")[$n − 4 = −8$]

#solutionbox[
−4
]
]


=== Model the Division Property of Equality

All of the equations we have solved so far have been of the form #math.equation(block: false, alt: "x plus a equals b")[$x + a = b$] or #math.equation(block: false, alt: "x minus a equals b .")[$x − a = b .$] We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.

We will model an #strong[equation] with envelopes and counters in .

#figure(figph[This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.], alt: "This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.", caption: none)

Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

To determine the number, separate the counters on the right side into #math.equation(block: false, alt: "2")[$2$] groups of the same size. So #math.equation(block: false, alt: "6")[$6$] counters divided into #math.equation(block: false, alt: "2")[$2$] groups means there must be #math.equation(block: false, alt: "3")[$3$] counters in each group (since #math.equation(block: false, alt: "6 divided by 2 equals 3 close parenthesis .")[$6 ÷ 2 = 3 \) .$]

What equation models the situation shown in ? There are two envelopes, and each contains #math.equation(block: false, alt: "x")[$x$] counters. Together, the two envelopes must contain a total of #math.equation(block: false, alt: "6")[$6$] counters. So the equation that models the situation is #math.equation(block: false, alt: "2 x equals 6 .")[$2 x = 6 .$]

#figure(figph[This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.], alt: "This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.", caption: none)

We can divide both sides of the equation by #math.equation(block: false, alt: "2")[$2$] as we did with the envelopes and counters.

#figure(figph[This figure has two rows. The first row has the equation 2x divided by 2 equals 6 divided by 2. The second row has the equation x equals 3.], alt: "This figure has two rows. The first row has the equation 2x divided by 2 equals 6 divided by 2. The second row has the equation x equals 3.", caption: none)

We found that each envelope contains #math.equation(block: false, alt: "3 counters.")[$"3 counters."$] Does this check? We know #math.equation(block: false, alt: "2 times 3 equals 6 ,")[$2 · 3 = 6 ,$] so it works. Three counters in each of two envelopes does equal six.

shows another example.

#figure(figph[This image has two columns. In the first column are three envelopes. In the second column there are four rows of  three blue circles. Underneath the image is the equation 3x equals 12.], alt: "This image has two columns. In the first column are three envelopes. In the second column there are four rows of three blue circles. Underneath the image is the equation 3x equals 12.", caption: none)

Now we have #math.equation(block: false, alt: "3")[$3$] identical envelopes and #math.equation(block: false, alt: "12 counters.")[$"12 counters."$] How many counters are in each envelope? We have to separate the #math.equation(block: false, alt: "12 counters")[$"12 counters"$] into #math.equation(block: false, alt: "3 groups.")[$"3 groups."$] Since #math.equation(block: false, alt: "12 divided by 3 equals 4 ,")[$12 ÷ 3 = 4 ,$] there must be #math.equation(block: false, alt: "4 counters")[$"4 counters"$] in each envelope.

#figure(figph[This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.], alt: "This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.", caption: none)

The equation that models the situation is #math.equation(block: false, alt: "3 x equals 12 .")[$3 x = 12 .$] We can divide both sides of the equation by #math.equation(block: false, alt: "3 .")[$3 .$]

#figure(figph[This image shows the equation 3x divided by 3 equals 12 divided by 3. Below this equation is the equation x equals 4.], alt: "This image shows the equation 3x divided by 3 equals 12 divided by 3. Below this equation is the equation x equals 4.", caption: none)

Does this check? It does because #math.equation(block: false, alt: "3 times 4 equals 12 .")[$3 · 4 = 12 .$]

#examplebox("Example 4")[][
Write an equation modeled by the envelopes and counters, and then solve it.

#figure(figph[This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.], alt: "This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.", caption: none)

#solutionbox[
There are #math.equation(block: false, alt: "4 envelopes,")[$"4 envelopes,"$] or #math.equation(block: false, alt: "4")[$4$] unknown values, on the left that match the #math.equation(block: false, alt: "8 counters")[$"8 counters"$] on the right. Let’s call the unknown quantity in the envelopes #math.equation(block: false, alt: "x .")[$x .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write the equation.], [#figure(figph[A simple algebraic equation '4x = 8' is displayed in black font on a white background.], alt: "A simple algebraic equation '4x = 8' is displayed in black font on a white background.", caption: none)]),
  [Divide both sides by 4.], [#figure(figph[An algebraic step demonstrating the division of both sides of the equation 4x = 8 by 4, to isolate x. This simplifies to x = 2.], alt: "An algebraic step demonstrating the division of both sides of the equation 4x = 8 by 4, to isolate x. This simplifies to x = 2.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression on a white background reads 'x = 2' in a dark, bold font.], alt: "A mathematical expression on a white background reads 'x = 2' in a dark, bold font.", caption: none)],
))

There are #math.equation(block: false, alt: "2 counters")[$"2 counters"$] in each envelope.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the equation modeled by the envelopes and counters. Then solve it. #linebreak() 
#figure(figph[This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles.], alt: "This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles.", caption: none)

#solutionbox[
4#emph[x] = 12; #emph[x] = 3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the equation modeled by the envelopes and counters. Then solve it. #linebreak() 
#figure(figph[This image has two columns. In the first column are three envelopes. In the second column there are six blue circles.], alt: "This image has two columns. In the first column are three envelopes. In the second column there are six blue circles.", caption: none)

#solutionbox[
3#emph[x] = 6; #emph[x] = 2
]
]


=== Solve Equations Using the Division Property of Equality

The previous examples lead to the #strong[Division Property of Equality]. When you divide both sides of an #strong[equation] by any nonzero number, you still have equality.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division Property of Equality]

#math.equation(block: true, alt: "For any numbers, a , b , c , and, c not equal to 0 ,; If, a equals b then, the fraction a over c equals the fraction b over c .")[$"For any numbers" & a , b , c , "and" & c ≠ 0 #h(.1em) , \ "If" & a = b #h(0.3em) "then" & frac(a, c) = frac(b, c) .$]
]

#examplebox("Example 5")[][
#math.equation(block: true, alt: "Solve: 7 x equals −49 .")[$"Solve:" #h(0.2em) 7 x = −49 .$]

#solutionbox[
To isolate #math.equation(block: false, alt: "x ,")[$x ,$] we need to undo multiplication. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing '7x = -49' in bold black characters on a white background. This is a linear equation where the variable x is multiplied by 7 and set equal to -49.], alt: "A mathematical equation is displayed, showing '7x = -49' in bold black characters on a white background. This is a linear equation where the variable x is multiplied by 7 and set equal to -49.", caption: none)]),
  [Divide each side by 7.], [#figure(figph[A mathematical equation showing the division of both sides by 7 to solve for x, represented as 7x/7 = -49/7. The number 7 in the denominator on both sides is highlighted in red.], alt: "A mathematical equation showing the division of both sides by 7 to solve for x, represented as 7x/7 = -49/7. The number 7 in the denominator on both sides is highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows 'x = -7' rendered in a bold, sans-serif style font against a clean white background. The variable 'x' is set equal to the negative integer '7'.], alt: "A mathematical equation shows 'x = -7' rendered in a bold, sans-serif style font against a clean white background. The variable 'x' is set equal to the negative integer '7'.", caption: none)],
))

Check the solution.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "7 x equals −49")[$7 x = −49 #h(1.35em)$]]),
  [Substitute −7 for x.], [#math.equation(block: false, alt: "7 open parenthesis −7 close parenthesis equals ? −49")[$7 ( −7 ) limits(=)^(?) −49 #h(1.35em)$]],
  [], [#math.equation(block: false, alt: "−49 equals −49 ✓")[$−49 = −49 ✓$]],
))

Therefore, #math.equation(block: false, alt: "−7")[$−7$] is the solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "8 a equals 56")[$8 a = 56$]

#solutionbox[
7
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "11 n equals 121")[$11 n = 121$]

#solutionbox[
11
]
]

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "−3 y equals 63 .")[$−3 y = 63 .$]

#solutionbox[
To isolate #math.equation(block: false, alt: "y ,")[$y ,$] we need to undo the multiplication. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation reads '-3y = 63' on a white background.], alt: "A mathematical equation reads '-3y = 63' on a white background.", caption: none)]),
  [Divide each side by −3.], [#figure(figph[A mathematical equation showing the division of -3y by -3 on the left side, which equals the division of 63 by -3 on the right side. The -3 in the denominator is highlighted in red on both sides.], alt: "A mathematical equation showing the division of -3y by -3 on the left side, which equals the division of 63 by -3 on the right side. The -3 in the denominator is highlighted in red on both sides.", caption: none)],
  [Simplify], [#figure(figph[The image displays a mathematical equation in black text on a white background, which states 'y = -21'.], alt: "The image displays a mathematical equation in black text on a white background, which states 'y = -21'.", caption: none)],
))

Check the solution.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−3 y equals 63")[$−3 y = 63 #h(1.35em)$]]),
  [Substitute −21 for y.], [#math.equation(block: false, alt: "−3 open parenthesis −21 close parenthesis equals ? 63")[$−3 ( −21 ) limits(=)^(?) 63 #h(1.35em)$]],
  [], [#math.equation(block: false, alt: "63 equals 63 ✓")[$63 = 63 ✓$]],
))

Since this is a true statement, #math.equation(block: false, alt: "y equals −21")[$y = −21$] is the solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "−8 p equals 96")[$−8 p = 96$]

#solutionbox[
−12
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "−12 m equals 108")[$−12 m = 108$]

#solutionbox[
−9
]
]


=== Translate to an Equation and Solve

In the past several examples, we were given an #strong[equation] containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

#examplebox("Example 7")[][
Translate and solve: five more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "−3 .")[$−3 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [five more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "−3")[$−3$]]),
  [Translate], [#math.equation(block: false, alt: "x plus 5 equals −3")[$x + 5 = −3$]],
  [Subtract #math.equation(block: false, alt: "5")[$5$] from both sides.], [#math.equation(block: false, alt: "x plus 5 minus 5 equals −3 minus 5")[$x + 5 − 5 = −3 − 5$]],
  [Simplify.], [#math.equation(block: false, alt: "x equals −8")[$x = "−8"$]],
))

Check the answer by substituting it into the original equation.

#math.equation(block: true, alt: "x plus 5 equals −3 ; −8 plus 5 equals ? −3 ; −3 equals −3 ✓")[$x + 5 = −3 #h(1.35em) \ −8 + 5 limits(=)^(?) −3 #h(1.35em) \ −3 = −3 ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

Seven more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "−2")[$−2$].

#solutionbox[
#emph[x] + 7 = −2; #emph[x] = −9
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

#math.equation(block: true, alt: "Eleven more than y is equal to 2.")[$"Eleven more than" #h(0.2em) y #h(0.2em) "is equal to 2."$]

#solutionbox[
#emph[y] + 11 = 2; #emph[y] = −9
]
]

#examplebox("Example 8")[][
Translate and solve: the difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "6")[$6$] is #math.equation(block: false, alt: "−10 .")[$−10 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [the difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "6")[$6$] is #math.equation(block: false, alt: "−10")[$−10$]]),
  [Translate.], [#math.equation(block: false, alt: "n minus 6 equals −10")[$n − 6 = −10$]],
  [Add #math.equation(block: false, alt: "6")[$6$] to each side.], [#math.equation(block: false, alt: "n minus 6 plus 6 equals −10 plus 6")[$n − 6 + 6 = −10 + 6$]],
  [Simplify.], [#math.equation(block: false, alt: "n equals −4")[$n = −4$]],
))

Check the answer by substituting it into the original equation.

#math.equation(block: true, alt: "n minus 6 equals −10 ; −4 minus 6 equals ? −10 ; −10 equals −10 ✓")[$#h(0.5em) n − 6 = −10 #h(1.35em) \ −4 − 6 limits(=)^(?) −10 #h(1.35em) \ −10 = −10 ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The difference of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "2")[$2$] is #math.equation(block: false, alt: "−4")[$−4$].

#solutionbox[
#emph[p] − 2 = −4; #emph[p] = −2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The difference of #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "7")[$7$] is #math.equation(block: false, alt: "−3")[$−3$].

#solutionbox[
#emph[q] − 7 = −3; #emph[q] = 4
]
]

#examplebox("Example 9")[][
Translate and solve: the number #math.equation(block: false, alt: "108")[$108$] is the product of #math.equation(block: false, alt: "−9")[$−9$] and #math.equation(block: false, alt: "y .")[$y .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [the number of #math.equation(block: false, alt: "108")[$108$] is the product of #math.equation(block: false, alt: "−9")[$−9$] and #math.equation(block: false, alt: "y")[$y$]]),
  [Translate.], [#math.equation(block: false, alt: "108 equals −9 y")[$108 = −9 y$]],
  [Divide by #math.equation(block: false, alt: "−9")[$−9$].], [#math.equation(block: false, alt: "the fraction 108 over −9 equals the fraction −9 y over −9")[$frac(108, −9) = frac(−9 y, −9)$]],
  [Simplify.], [#math.equation(block: false, alt: "−12 equals y")[$−12 = y$]],
))

Check the answer by substituting it into the original equation.

#math.equation(block: true, alt: "108 equals −9 y; 108 equals ? −9 open parenthesis −12 close parenthesis; 108 equals 108 ✓")[$108 = −9 y \ 108 limits(=)^(?) −9 ( −12 ) \ 108 = 108 ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The number #math.equation(block: false, alt: "132")[$132$] is the product of #math.equation(block: false, alt: "−12")[$−12$] and #math.equation(block: false, alt: "y")[$y$].

#solutionbox[
132 = −12#emph[y]; #emph[y] = −11
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The number #math.equation(block: false, alt: "117")[$117$] is the product of #math.equation(block: false, alt: "−13")[$−13$] and #math.equation(block: false, alt: "z")[$z$].

#solutionbox[
117 = −13#emph[z]; #emph[z] = −9
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24onestepaddsub")[One-Step Equations With Adding Or Subtracting]
- #link("https://openstaxcollege.org/l/24onestepmuldiv")[One-Step Equations With Multiplying Or Dividing]
]


=== Key Concepts

- #strong[How to determine whether a number is a solution to an equation.]
- Step 1. Substitute the number for the variable in the equation.
- Step 2. Simplify the expressions on both sides of the equation.
- Step 3. Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
- #strong[Properties of Equalities]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Subtraction Property of Equality], [Addition Property of Equality]),
  [#math.equation(block: false, alt: "For any numbers a , b , c ,")[$"For any numbers" #h(0.2em) a , b , c ,$] #linebreak() #math.equation(block: false, alt: "if a equals b then a minus c equals b minus c .")[$"if" #h(0.2em) a = b #h(0.2em) "then" #h(0.2em) a − c = b − c .$]], [#math.equation(block: false, alt: "For any numbers a , b , c ,")[$"For any numbers" #h(0.2em) a , b , c ,$] #linebreak() #math.equation(block: false, alt: "if a equals b then a plus c equals b plus c .")[$"if" #h(0.2em) a = b #h(0.2em) "then" #h(0.2em) a + c = b + c .$]],
))
- #strong[Division Property of Equality]


- For any numbers #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] and #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$] #linebreak()  If #math.equation(block: false, alt: "a equals b")[$a = b$], then

#math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$].


=== Section Exercises


==== Practice Makes Perfect

#strong[Determine Whether a Number is a Solution of an Equation]

In the following exercises, determine whether each number is a solution of the given equation.

#math.equation(block: true, alt: "4 x minus 2 equals 6")[$4 x − 2 = 6$]

+ ⓐ #math.equation(block: false, alt: "x equals −2")[$#h(0.2em) x = −2$]
+ ⓑ #math.equation(block: false, alt: "x equals −1")[$#h(0.2em) x = −1$]
+ ⓒ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2$]

#solutionbox[
+ ⓐ no
+ ⓑ no
+ ⓒ yes
]

#math.equation(block: true, alt: "4 y minus 10 equals −14")[$4 y − 10 = −14$]

+ ⓐ #math.equation(block: false, alt: "y equals −6")[$#h(0.2em) y = −6$]
+ ⓑ #math.equation(block: false, alt: "y equals −1")[$#h(0.2em) y = −1$]
+ ⓒ #math.equation(block: false, alt: "y equals 1")[$#h(0.2em) y = 1$]

#math.equation(block: true, alt: "9 a plus 27 equals −63")[$9 a + 27 = −63$]

+ ⓐ #math.equation(block: false, alt: "a equals 6")[$#h(0.2em) a = 6$]
+ ⓑ #math.equation(block: false, alt: "a equals −6")[$#h(0.2em) a = −6$]
+ ⓒ #math.equation(block: false, alt: "a equals −10")[$#h(0.2em) a = −10$]

#solutionbox[
+ ⓐ no
+ ⓑ no
+ ⓒ yes
]

#math.equation(block: true, alt: "7 c plus 42 equals −56")[$7 c + 42 = −56$]

+ ⓐ #math.equation(block: false, alt: "c equals 2")[$c = 2$]
+ ⓑ #math.equation(block: false, alt: "c equals −2")[$c = −2$]
+ ⓒ #math.equation(block: false, alt: "c equals −14")[$c = −14$]

#strong[Solve Equations Using the Addition and Subtraction Properties of Equality]

In the following exercises, solve for the unknown.

#math.equation(block: true, alt: "n plus 12 equals 5")[$n + 12 = 5$]

#solutionbox[
#emph[n] = −7
]

#math.equation(block: true, alt: "m plus 16 equals 2")[$m + 16 = 2$]

#math.equation(block: true, alt: "p plus 9 equals −8")[$p + 9 = −8$]

#solutionbox[
#emph[p] = −17
]

#math.equation(block: true, alt: "q plus 5 equals −6")[$q + 5 = −6$]

#math.equation(block: true, alt: "u minus 3 equals −7")[$u − 3 = −7$]

#solutionbox[
#emph[u] = −4
]

#math.equation(block: true, alt: "v minus 7 equals −8")[$v − 7 = −8$]

#math.equation(block: true, alt: "h minus 10 equals −4")[$h − 10 = −4$]

#solutionbox[
#emph[h] = 6
]

#math.equation(block: true, alt: "k minus 9 equals −5")[$k − 9 = −5$]

#math.equation(block: true, alt: "x plus open parenthesis −2 close parenthesis equals −18")[$x + ( −2 ) = −18$]

#solutionbox[
#emph[x] = −16
]

#math.equation(block: true, alt: "y plus open parenthesis −3 close parenthesis equals −10")[$y + ( −3 ) = −10$]

#math.equation(block: true, alt: "r minus open parenthesis −5 close parenthesis equals −9")[$r − ( −5 ) = −9$]

#solutionbox[
#emph[r] = −14
]

#math.equation(block: true, alt: "s minus open parenthesis −2 close parenthesis equals −11")[$s − ( −2 ) = −11$]

#strong[Model the Division Property of Equality]

In the following exercises, write the equation modeled by the envelopes and counters and then solve it.

#figure(figph[Three light blue envelopes are stacked vertically on the left side of a white background, while six light yellow circular tokens are arranged in two columns of three on the right.], alt: "Three light blue envelopes are stacked vertically on the left side of a white background, while six light yellow circular tokens are arranged in two columns of three on the right.", caption: none)

#solutionbox[
3#emph[x] = 6; #emph[x] = 2
]

#figure(figph[Two light blue envelopes are paired with ten light yellow circles arranged in two columns of five on a white background.], alt: "Two light blue envelopes are paired with ten light yellow circles arranged in two columns of five on a white background.", caption: none)

#figure(figph[Two light blue envelopes are paired with eight light yellow circles arranged in two columns of four on a white background.], alt: "Two light blue envelopes are paired with eight light yellow circles arranged in two columns of four on a white background.", caption: none)

#solutionbox[
2#emph[x] = 8; #emph[x] = 4
]

#figure(figph[Three light blue envelopes are displayed on the left, while nine yellow-rimmed circles are arranged in a 3x3 grid on the right, all on a white background.], alt: "Three light blue envelopes are displayed on the left, while nine yellow-rimmed circles are arranged in a 3x3 grid on the right, all on a white background.", caption: none)

#strong[Solve Equations Using the Division Property of Equality]

In the following exercises, solve each equation using the division property of equality and check the solution.

#math.equation(block: true, alt: "5 x equals 45")[$5 x = 45$]

#solutionbox[
#emph[x] = 9
]

#math.equation(block: true, alt: "4 p equals 64")[$4 p = 64$]

#math.equation(block: true, alt: "−7 c equals 56")[$−7 c = 56$]

#solutionbox[
#emph[c] = −8
]

#math.equation(block: true, alt: "−9 x equals 54")[$−9 x = 54$]

#math.equation(block: true, alt: "−14 p equals −42")[$−14 p = −42$]

#solutionbox[
#emph[p] = 3
]

#math.equation(block: true, alt: "−8 m equals −40")[$−8 m = −40$]

#math.equation(block: true, alt: "−120 equals 10 q")[$−120 = 10 q$]

#solutionbox[
#emph[q] = −12
]

#math.equation(block: true, alt: "−75 equals 15 y")[$−75 = 15 y$]

#math.equation(block: true, alt: "24 x equals 480")[$24 x = 480$]

#solutionbox[
#emph[x] = 20
]

#math.equation(block: true, alt: "18 n equals 540")[$18 n = 540$]

#math.equation(block: true, alt: "−3 z equals 0")[$−3 z = 0$]

#solutionbox[
#emph[z] = 0
]

#math.equation(block: true, alt: "4 u equals 0")[$4 u = 0$]

#strong[Translate to an Equation and Solve]

In the following exercises, translate and solve.

Four more than #math.equation(block: false, alt: "n")[$n$] is equal to 1.

#solutionbox[
#emph[n] + 4 = 1; #emph[n] = −3
]

Nine more than #math.equation(block: false, alt: "m")[$m$] is equal to 5.

The sum of eight and #math.equation(block: false, alt: "p")[$p$] is #math.equation(block: false, alt: "−3")[$−3$].

#solutionbox[
8 + #emph[p] = −3; #emph[p] = −11
]

The sum of two and #math.equation(block: false, alt: "q")[$q$] is #math.equation(block: false, alt: "−7")[$−7$].

The difference of #math.equation(block: false, alt: "a")[$a$] and three is #math.equation(block: false, alt: "−14")[$−14$].

#solutionbox[
#emph[a] − 3 = −14; #emph[a] = −11
]

The difference of #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "5")[$5$] is #math.equation(block: false, alt: "−2")[$−2$].

The number −42 is the product of −7 and #math.equation(block: false, alt: "x")[$x$].

#solutionbox[
−42 = −7#emph[x]; #emph[x] = 6
]

The number −54 is the product of −9 and #math.equation(block: false, alt: "y")[$y$].

The product of -15 and #math.equation(block: false, alt: "f")[$f$] is 75.

#solutionbox[
−15#emph[f] = 75; #emph[f] = −5
]

The product of −18 and #math.equation(block: false, alt: "g")[$g$] is 36.

−6 plus #math.equation(block: false, alt: "c")[$c$] is equal to 4.

#solutionbox[
−6 + #emph[c] = 4; #emph[c] = 10
]

−2 plus #math.equation(block: false, alt: "d")[$d$] is equal to 1.

Nine less than #math.equation(block: false, alt: "m")[$m$] is −4.

#solutionbox[
#emph[m] − 9 = −4; #emph[m] = 5
]

Thirteen less than #math.equation(block: false, alt: "n")[$n$] is #math.equation(block: false, alt: "−10")[$−10$].

#strong[Mixed Practice]

In the following exercises, solve.

+ ⓐ #math.equation(block: false, alt: "x plus 2 equals 10")[$#h(0.2em) x + 2 = 10 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "2 x equals 10")[$#h(0.2em) 2 x = 10$]

#solutionbox[
+ ⓐ #emph[x] = 8
+ ⓑ #emph[x] = 5
]

+ ⓐ #math.equation(block: false, alt: "y plus 6 equals 12")[$#h(0.2em) y + 6 = 12 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "6 y equals 12")[$#h(0.2em) 6 y = 12$]

+ ⓐ #math.equation(block: false, alt: "−3 p equals 27")[$#h(0.2em) −3 p = 27 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "p minus 3 equals 27")[$#h(0.2em) p − 3 = 27$]

#solutionbox[
+ ⓐ #emph[p] = −9
+ ⓑ #emph[p] = 30
]

+ ⓐ #math.equation(block: false, alt: "−2 q equals 34")[$#h(0.2em) −2 q = 34 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "q minus 2 equals 34")[$#h(0.2em) q − 2 = 34$]

#math.equation(block: true, alt: "a minus 4 equals 16")[$a − 4 = 16$]

#solutionbox[
#emph[a] = 20
]

#math.equation(block: true, alt: "b minus 1 equals 11")[$b − 1 = 11$]

#math.equation(block: true, alt: "−8 m equals −56")[$−8 m = −56$]

#solutionbox[
#emph[m] = 7
]

#math.equation(block: true, alt: "−6 n equals −48")[$−6 n = −48$]

#math.equation(block: true, alt: "−39 equals u plus 13")[$−39 = u + 13$]

#solutionbox[
#emph[u] = −52
]

#math.equation(block: true, alt: "−100 equals v plus 25")[$−100 = v + 25$]

#math.equation(block: true, alt: "11 r equals −99")[$11 r = −99$]

#solutionbox[
#emph[r] = −9
]

#math.equation(block: true, alt: "15 s equals −300")[$15 s = −300$]

#math.equation(block: true, alt: "100 equals 20 d")[$100 = 20 d$]

#solutionbox[
#emph[d] = 5
]

#math.equation(block: true, alt: "250 equals 25 n")[$250 = 25 n$]

#math.equation(block: true, alt: "−49 equals x minus 7")[$−49 = x − 7$]

#solutionbox[
#emph[x] = −42
]

#math.equation(block: true, alt: "64 equals y minus 4")[$64 = y − 4$]


==== Everyday Math

#strong[Cookie packaging] A package of #math.equation(block: false, alt: "51 cookies")[$"51 cookies"$] has #math.equation(block: false, alt: "3")[$3$] equal rows of cookies. Find the number of cookies in each row, #math.equation(block: false, alt: "c ,")[$c ,$] by solving the equation #math.equation(block: false, alt: "3 c equals 51 .")[$3 c = 51 .$]

#solutionbox[
17 cookies
]

#strong[Kindergarten class] Connie’s kindergarten class has #math.equation(block: false, alt: "24 children.")[$"24 children."$] She wants them to get into #math.equation(block: false, alt: "4")[$4$] equal groups. Find the number of children in each group, #math.equation(block: false, alt: "g ,")[$g ,$] by solving the equation #math.equation(block: false, alt: "4 g equals 24 .")[$4 g = 24 .$]


==== Writing Exercises

Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation #math.equation(block: false, alt: "3 x equals 15 ?")[$3 x = 15 ?$] Explain why or why not.

#solutionbox[
Sample answer: It is helpful because it shows how the counters can be divided among the envelopes.
]

Suppose you are using envelopes and counters to model solving the equations #math.equation(block: false, alt: "x plus 4 equals 12")[$x + 4 = 12$] and #math.equation(block: false, alt: "4 x equals 12 .")[$4 x = 12 .$] Explain how you would solve each equation.

Frida started to solve the equation #math.equation(block: false, alt: "−3 x equals 36")[$−3 x = 36$] by adding #math.equation(block: false, alt: "3")[$3$] to both sides. Explain why Frida’s method will not solve the equation.

#solutionbox[
Sample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition.
]

Raoul started to solve the equation #math.equation(block: false, alt: "4 y equals 40")[$4 y = 40$] by subtracting #math.equation(block: false, alt: "4")[$4$] from both sides. Explain why Raoul’s method will not solve the equation.


==== Self Check

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

#figure(figph[A self-assessment checklist for math skills, asking users to rate their ability to solve equations with integers and understand properties of equality using options like 'Confidently', 'With some help', or 'No-I don't get it!'.], alt: "A self-assessment checklist for math skills, asking users to rate their ability to solve equations with integers and understand properties of equality using options like 'Confidently', 'With some help', or 'No-I don't get it!'.", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


=== Chapter Review Exercises


==== Introduction to Integers

#strong[Locate Positive and Negative Numbers on the Number Line]

In the following exercises, locate and label the integer on the number line.

#math.equation(block: true, alt: "5")[$5$]

#solutionbox[
#figure(figph[This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at 5.], alt: "This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at 5.", caption: none)
]

#math.equation(block: true, alt: "−5")[$−5$]

#math.equation(block: true, alt: "−3")[$−3$]

#solutionbox[
#figure(figph[This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 3.], alt: "This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 3.", caption: none)
]

#math.equation(block: true, alt: "3")[$3$]

#math.equation(block: true, alt: "−8")[$−8$]

#solutionbox[
#figure(figph[This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 8.], alt: "This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 8.", caption: none)
]

#math.equation(block: true, alt: "−7")[$−7$]

#strong[Order Positive and Negative Numbers]

In the following exercises, order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than")[$<$] or #math.equation(block: false, alt: ">.")[$">."$]

#math.equation(block: true, alt: "4 __ 8")[$4 "__" 8$]

#solutionbox[
\<
]

#math.equation(block: true, alt: "−6 __ 3")[$−6 "__" 3$]

#math.equation(block: true, alt: "−5 __ −10")[$−5 "__" −10$]

#solutionbox[
\>
]

#math.equation(block: true, alt: "−9 __ −4")[$−9 "__" −4$]

#math.equation(block: true, alt: "2 __ −7")[$2 "__" −7$]

#solutionbox[
\>
]

#math.equation(block: true, alt: "−3 __ 1")[$−3 "__" 1$]

#strong[Find Opposites]

In the following exercises, find the opposite of each number.

#math.equation(block: true, alt: "6")[$6$]

#solutionbox[
−6
]

#math.equation(block: true, alt: "−2")[$−2$]

#math.equation(block: true, alt: "−4")[$−4$]

#solutionbox[
4
]

#math.equation(block: true, alt: "3")[$3$]

In the following exercises, simplify.

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis 8 close parenthesis")[$#h(0.2em) − ( 8 ) #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "minus open parenthesis −8 close parenthesis")[$#h(0.2em) − ( −8 )$]

#solutionbox[
+ ⓐ −8
+ ⓑ 8
]

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis 9 close parenthesis")[$#h(0.2em) − ( 9 ) #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "minus open parenthesis −9 close parenthesis")[$#h(0.2em) − ( −9 )$]

In the following exercises, evaluate.

#math.equation(block: true, alt: "minus x , when")[$− x , #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals 32")[$#h(0.2em) x = 32 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −32")[$#h(0.2em) x = −32$]

#solutionbox[
+ ⓐ −32
+ ⓑ 32
]

#math.equation(block: true, alt: "minus n , when")[$− n , #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "n equals 20")[$#h(0.2em) n = 20 #h(0.6em)$]
+ ⓐ #math.equation(block: false, alt: "n equals −20")[$#h(0.2em) n = −20$]

#strong[Simplify Absolute Values]

In the following exercises, simplify.

#math.equation(block: true, alt: "| −21 |")[$| −21 |$]

#solutionbox[
21
]

#math.equation(block: true, alt: "| −42 |")[$| −42 |$]

#math.equation(block: true, alt: "| 36 |")[$| 36 |$]

#solutionbox[
36
]

#math.equation(block: true, alt: "minus | 15 |")[$− | 15 |$]

#math.equation(block: true, alt: "| 0 |")[$| 0 |$]

#solutionbox[
0
]

#math.equation(block: true, alt: "minus | −75 |")[$− | −75 |$]

In the following exercises, evaluate.

#math.equation(block: true, alt: "| x | when x equals −14")[$| x | #h(0.2em) "when" #h(0.2em) x = −14$]

#solutionbox[
14
]

#math.equation(block: true, alt: "minus | r | when r equals 27")[$− | r | #h(0.2em) "when" #h(0.2em) r = 27$]

#math.equation(block: true, alt: "minus | minus y | when y equals 33")[$− | − y | #h(0.2em) "when" #h(0.2em) y = 33$]

#solutionbox[
−33
]

#math.equation(block: true, alt: "| −n | when n equals −4")[$| "−n" | #h(0.2em) "when" #h(0.2em) n = −4$]

In the following exercises, fill in #math.equation(block: false, alt: "< , > , or equals")[$"<" , ">" , "or" =$] for each of the following pairs of numbers.

#math.equation(block: true, alt: "minus | −4 | __ 4")[$− | −4 | "__" 4$]

#solutionbox[
\<
]

#math.equation(block: true, alt: "−2 __ | −2 |")[$−2 "__" | −2 |$]

#math.equation(block: true, alt: "minus | −6 | __ −6")[$− | −6 | "__" −6$]

#solutionbox[
=
]

#math.equation(block: true, alt: "minus | −9 | __ | −9 |")[$− | −9 | "__" | −9 |$]

In the following exercises, simplify.

#math.equation(block: true, alt: "minus open parenthesis −55 close parenthesis and minus | −55 |")[$− ( −55 ) #h(0.2em) "and" #h(0.2em) − | −55 |$]

#solutionbox[
55; −55
]

#math.equation(block: true, alt: "minus open parenthesis −48 close parenthesis and minus | −48 |")[$− ( −48 ) #h(0.2em) "and" #h(0.2em) − | −48 |$]

#math.equation(block: true, alt: "| 12 minus 5 |")[$| 12 − 5 |$]

#solutionbox[
7
]

#math.equation(block: true, alt: "| 9 plus 7 |")[$| 9 + 7 |$]

#math.equation(block: true, alt: "6 | −9 |")[$6 | −9 |$]

#solutionbox[
54
]

#math.equation(block: true, alt: "| 14 −8 | minus | −2 |")[$| 14 −8 | − | −2 |$]

#math.equation(block: true, alt: "| 9 minus 3 | minus | 5 minus 12 |")[$| 9 − 3 | − | 5 − 12 |$]

#solutionbox[
−1
]

#math.equation(block: true, alt: "5 plus 4 | 15 minus 3 |")[$5 + 4 | 15 − 3 |$]

#strong[Translate Phrases to Expressions with Integers]

In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

the opposite of #math.equation(block: false, alt: "16")[$16$]

#solutionbox[
−16
]

the opposite of #math.equation(block: false, alt: "−8")[$−8$]

negative #math.equation(block: false, alt: "3")[$3$]

#solutionbox[
−3
]

#math.equation(block: false, alt: "19")[$19$] minus negative #math.equation(block: false, alt: "12")[$12$]

a temperature of #math.equation(block: false, alt: "10")[$10$] below zero

#solutionbox[
−10°
]

an elevation of #math.equation(block: false, alt: "85 feet")[$"85 feet"$] below sea level


==== Add Integers

#strong[Model Addition of Integers]

In the following exercises, model the following to find the sum.

#math.equation(block: true, alt: "3 plus 7")[$3 + 7$]

#solutionbox[
10
]

#math.equation(block: true, alt: "−2 plus 6")[$−2 + 6$]

#math.equation(block: true, alt: "5 plus open parenthesis −4 close parenthesis")[$5 + ( −4 )$]

#solutionbox[
1
]

#math.equation(block: true, alt: "−3 plus open parenthesis −6 close parenthesis")[$−3 + ( −6 )$]

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "14 plus 82")[$14 + 82$]

#solutionbox[
96
]

#math.equation(block: true, alt: "−33 plus open parenthesis −67 close parenthesis")[$−33 + ( −67 )$]

#math.equation(block: true, alt: "−75 plus 25")[$−75 + 25$]

#solutionbox[
−50
]

#math.equation(block: true, alt: "54 plus open parenthesis −28 close parenthesis")[$54 + ( −28 )$]

#math.equation(block: true, alt: "11 plus open parenthesis −15 close parenthesis plus 3")[$11 + ( −15 ) + 3$]

#solutionbox[
−1
]

#math.equation(block: true, alt: "−19 plus open parenthesis −42 close parenthesis plus 12")[$−19 + ( −42 ) + 12$]

#math.equation(block: true, alt: "−3 plus 6 open parenthesis −1 plus 5 close parenthesis")[$−3 + 6 ( −1 + 5 )$]

#solutionbox[
21
]

#math.equation(block: true, alt: "10 plus 4 open parenthesis −3 plus 7 close parenthesis")[$10 + 4 ( −3 + 7 )$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: true, alt: "n plus 4 when")[$n + 4 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "n equals −1")[$#h(0.2em) n = −1 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "n equals −20")[$#h(0.2em) n = −20$]

#solutionbox[
+ ⓐ 3
+ ⓑ −16
]

#math.equation(block: true, alt: "x plus open parenthesis −9 close parenthesis when")[$x + ( −9 ) #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −3")[$#h(0.2em) x = −3$]

#math.equation(block: true, alt: "open parenthesis x plus y close parenthesis cubed when x equals −4 , y equals 1")[$attach(( x + y ), t: 3) #h(0.2em) "when" #h(0.2em) x = −4 , y = 1$]

#solutionbox[
−27
]

#math.equation(block: true, alt: "open parenthesis u plus v close parenthesis squared when u equals −4 , v equals 11")[$attach(( u + v ), t: 2) #h(0.2em) "when" #h(0.2em) u = −4 , v = 11$]

#strong[Translate Word Phrases to Algebraic Expressions]

In the following exercises, translate each phrase into an algebraic expression and then simplify.

#math.equation(block: true, alt: "the sum of −8 and 2")[$"the sum of −8 and 2"$]

#solutionbox[
−8 + 2 = −6
]

#math.equation(block: true, alt: "4 more than −12")[$"4 more than −12"$]

#math.equation(block: true, alt: "10 more than the sum of −5 and −6")[$"10 more than the sum of −5 and −6"$]

#solutionbox[
10 + \[−5 + (−6)\] = −1
]

#math.equation(block: true, alt: "the sum of 3 and −5 , increased by 18")[$"the sum of" #h(0.2em) 3 #h(0.2em) "and" #h(0.2em) −5 , #h(0.2em) "increased by 18"$]

#strong[Add Integers in Applications]

In the following exercises, solve.

#strong[Temperature] On Monday, the high temperature in Denver was #math.equation(block: false, alt: "−4 degrees.")[$"−4 degrees."$] Tuesday’s high temperature was #math.equation(block: false, alt: "20 degrees")[$"20 degrees"$] more. What was the high temperature on Tuesday?

#solutionbox[
16 degrees
]

#strong[Credit] Frida owed #math.equation(block: false, alt: "$75")[$"$75"$] on her credit card. Then she charged #math.equation(block: false, alt: "$21")[$"$21"$] more. What was her new balance?


==== Subtract Integers

#strong[Model Subtraction of Integers]

In the following exercises, model the following.

#math.equation(block: true, alt: "6 minus 1")[$6 − 1$]

#solutionbox[
#figure(figph[This figure is a row of 6 light pink circles, representing positive counters. The first one is circled.], alt: "This figure is a row of 6 light pink circles, representing positive counters. The first one is circled.", caption: none)

 #linebreak() 5
]

#math.equation(block: true, alt: "−4 minus open parenthesis −3 close parenthesis")[$−4 − ( −3 )$]

#math.equation(block: true, alt: "2 minus open parenthesis −5 close parenthesis")[$2 − ( −5 )$]

#solutionbox[
#figure(figph[This figure shows 2 rows. The first row shows 7 light pink circles, representing positive counters. The second row shows 5 dark pink circles, representing negative counters. The entire second row is circled.], alt: "This figure shows 2 rows. The first row shows 7 light pink circles, representing positive counters. The second row shows 5 dark pink circles, representing negative counters. The entire second row is circled.", caption: none)

 #linebreak() 7
]

#math.equation(block: true, alt: "−1 minus 4")[$−1 − 4$]

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "24 minus 16")[$24 − 16$]

#solutionbox[
8
]

#math.equation(block: true, alt: "19 minus open parenthesis −9 close parenthesis")[$19 − ( −9 )$]

#math.equation(block: true, alt: "−31 minus 7")[$−31 − 7$]

#solutionbox[
−38
]

#math.equation(block: true, alt: "−40 minus open parenthesis −11 close parenthesis")[$−40 − ( −11 )$]

#math.equation(block: true, alt: "−52 minus open parenthesis −17 close parenthesis minus 23")[$−52 − ( −17 ) − 23$]

#solutionbox[
−58
]

#math.equation(block: true, alt: "25 minus open parenthesis −3 minus 9 close parenthesis")[$25 − ( −3 − 9 )$]

#math.equation(block: true, alt: "open parenthesis 1 minus 7 close parenthesis minus open parenthesis 3 minus 8 close parenthesis")[$( 1 − 7 ) − ( 3 − 8 )$]

#solutionbox[
−1
]

#math.equation(block: true, alt: "3 squared minus 7 squared")[$3^(2) − 7^(2)$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: true, alt: "x minus 7 when")[$x − 7 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals 5")[$#h(0.2em) x = 5 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −4")[$#h(0.2em) x = −4$]

#solutionbox[
+ ⓐ −2
+ ⓑ −11
]

#math.equation(block: true, alt: "10 minus y when")[$10 − y #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "y equals 15")[$#h(0.2em) y = 15 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "y equals −16")[$#h(0.2em) y = −16$]

#math.equation(block: true, alt: "2 n squared minus n plus 5 when n equals −4")[$2 n^(2) − n + 5 #h(0.2em) "when" #h(0.2em) n = −4$]

#solutionbox[
41
]

#math.equation(block: true, alt: "−15 minus 3 u squared when u equals −5")[$−15 − 3 u^(2) #h(0.2em) "when" #h(0.2em) u = −5$]

#strong[Translate Phrases to Algebraic Expressions]

In the following exercises, translate each phrase into an algebraic expression and then simplify.

the difference of #math.equation(block: false, alt: "−12 and 5")[$−12 #h(0.2em) "and" #h(0.2em) 5$]

#solutionbox[
−12 − 5 = −17
]

subtract #math.equation(block: false, alt: "23")[$23$] from #math.equation(block: false, alt: "−50")[$−50$]

#strong[Subtract Integers in Applications]

In the following exercises, solve the given applications.

#strong[Temperature] One morning the temperature in Bangor, Maine was #math.equation(block: false, alt: "18 degrees.")[$"18 degrees."$] By afternoon, it had dropped #math.equation(block: false, alt: "20 degrees.")[$"20 degrees."$] What was the afternoon temperature?

#solutionbox[
−2 degrees
]

#strong[Temperature] On January 4, the high temperature in Laredo, Texas was #math.equation(block: false, alt: "78 degrees,")[$"78 degrees,"$] and the high in Houlton, Maine was #math.equation(block: false, alt: "−28 degrees .")[$−28 #h(0.2em) "degrees" .$] What was the difference in temperature of Laredo and Houlton?


==== Multiply and Divide Integers

#strong[Multiply Integers]

In the following exercises, multiply.

#math.equation(block: true, alt: "−9 times 4")[$−9 ⋅ 4$]

#solutionbox[
−36
]

#math.equation(block: true, alt: "5 open parenthesis −7 close parenthesis")[$5 ( −7 )$]

#math.equation(block: true, alt: "open parenthesis −11 close parenthesis open parenthesis −11 close parenthesis")[$( −11 ) ( −11 )$]

#solutionbox[
121
]

#math.equation(block: true, alt: "−1 times 6")[$−1 ⋅ 6$]

#strong[Divide Integers]

In the following exercises, divide.

#math.equation(block: true, alt: "56 divided by open parenthesis −8 close parenthesis")[$56 ÷ ( −8 )$]

#solutionbox[
−7
]

#math.equation(block: true, alt: "−120 divided by open parenthesis −6 close parenthesis")[$−120 ÷ ( −6 )$]

#math.equation(block: true, alt: "−96 divided by 12")[$−96 ÷ 12$]

#solutionbox[
−8
]

#math.equation(block: true, alt: "96 divided by open parenthesis −16 close parenthesis")[$96 ÷ ( −16 )$]

#math.equation(block: true, alt: "45 divided by open parenthesis −1 close parenthesis")[$45 ÷ ( −1 )$]

#solutionbox[
−45
]

#math.equation(block: true, alt: "−162 divided by open parenthesis −1 close parenthesis")[$−162 ÷ ( −1 )$]

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "5 open parenthesis −9 close parenthesis minus 3 open parenthesis −12 close parenthesis")[$5 ( −9 ) − 3 ( −12 )$]

#solutionbox[
−9
]

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis to the power 5")[$attach(( −2 ), t: 5)$]

#math.equation(block: true, alt: "minus 3 to the power 4")[$− 3^(4)$]

#solutionbox[
−81
]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis open parenthesis 4 close parenthesis open parenthesis −5 close parenthesis open parenthesis −6 close parenthesis")[$( −3 ) ( 4 ) ( −5 ) ( −6 )$]

#math.equation(block: true, alt: "42 minus 4 open parenthesis 6 minus 9 close parenthesis")[$42 − 4 ( 6 − 9 )$]

#solutionbox[
54
]

#math.equation(block: true, alt: "open parenthesis 8 minus 15 close parenthesis open parenthesis 9 minus 3 close parenthesis")[$( 8 − 15 ) ( 9 − 3 )$]

#math.equation(block: true, alt: "−2 open parenthesis −18 close parenthesis divided by 9")[$−2 ( −18 ) ÷ 9$]

#solutionbox[
4
]

#math.equation(block: true, alt: "45 divided by open parenthesis −3 close parenthesis minus 12")[$45 ÷ ( −3 ) − 12$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: true, alt: "7 x minus 3 when x equals −9")[$7 x − 3 #h(0.2em) "when" #h(0.2em) x = −9$]

#solutionbox[
−66
]

#math.equation(block: true, alt: "16 minus 2 n when n equals −8")[$16 − 2 n #h(0.2em) "when" #h(0.2em) n = −8$]

#math.equation(block: true, alt: "5 a plus 8 b when a equals −2 , b equals −6")[$5 a + 8 b #h(0.2em) "when" #h(0.2em) a = −2 , b = −6$]

#solutionbox[
−58
]

#math.equation(block: true, alt: "x squared plus 5 x plus 4 when x equals −3")[$x^(2) + 5 x + 4 #h(0.2em) "when" #h(0.2em) x = −3$]

#strong[Translate Word Phrases to Algebraic Expressions]

In the following exercises, translate to an algebraic expression and simplify if possible.

the product of #math.equation(block: false, alt: "−12")[$−12$] and #math.equation(block: false, alt: "6")[$6$]

#solutionbox[
−12(6) = −72
]

the quotient of #math.equation(block: false, alt: "3")[$3$] and the sum of #math.equation(block: false, alt: "−7")[$−7$] and #math.equation(block: false, alt: "s")[$s$]


==== Solve Equations using Integers; The Division Property of Equality

#strong[Determine Whether a Number is a Solution of an Equation]

In the following exercises, determine whether each number is a solution of the given equation.

#math.equation(block: true, alt: "5 x minus 10 equals −35")[$5 x − 10 = −35$]

+ ⓐ #math.equation(block: false, alt: "x equals −9")[$#h(0.2em) x = −9$]
+ ⓑ #math.equation(block: false, alt: "x equals −5")[$#h(0.2em) x = −5$]
+ ⓒ #math.equation(block: false, alt: "x equals 5")[$#h(0.2em) x = 5$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
+ ⓒ no
]

#math.equation(block: true, alt: "8 u plus 24 equals −32")[$8 u + 24 = −32$]

+ ⓐ #math.equation(block: false, alt: "u equals −7")[$#h(0.2em) u = −7$]
+ ⓑ #math.equation(block: false, alt: "u equals −1")[$#h(0.2em) u = −1$]
+ ⓒ #math.equation(block: false, alt: "u equals 7")[$#h(0.2em) u = 7$]

#strong[Using the Addition and Subtraction Properties of Equality]

In the following exercises, solve.

#math.equation(block: true, alt: "a plus 14 equals 2")[$a + 14 = 2$]

#solutionbox[
−12
]

#math.equation(block: true, alt: "b minus 9 equals −15")[$b − 9 = −15$]

#math.equation(block: true, alt: "c plus open parenthesis −10 close parenthesis equals −17")[$c + ( −10 ) = −17$]

#solutionbox[
−7
]

#math.equation(block: true, alt: "d minus open parenthesis −6 close parenthesis equals −26")[$d − ( −6 ) = −26$]

#strong[Model the Division Property of Equality]

In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.

#figure(figph[This image has two columns. In the first column there are three envelopes. In the second column there are two vertical rows. The first row includes five blue circles, the second row includes four blue circles.], alt: "This image has two columns. In the first column there are three envelopes. In the second column there are two vertical rows. The first row includes five blue circles, the second row includes four blue circles.", caption: none)

#solutionbox[
3#emph[x] = 9; #emph[x] = 3
]

#figure(figph[This figure has two columns. In the first column there are  two envelopes. In the second column there are two vertical rows, each includes four blue circles.], alt: "This figure has two columns. In the first column there are two envelopes. In the second column there are two vertical rows, each includes four blue circles.", caption: none)

#strong[Solve Equations Using the Division Property of Equality]

In the following exercises, solve each equation using the division property of equality and check the solution.

#math.equation(block: true, alt: "8 p equals 72")[$8 p = 72$]

#solutionbox[
9
]

#math.equation(block: true, alt: "−12 q equals 48")[$−12 q = 48$]

#math.equation(block: true, alt: "−16 r equals −64")[$−16 r = −64$]

#solutionbox[
4
]

#math.equation(block: true, alt: "−5 s equals −100")[$−5 s = −100$]

#strong[Translate to an Equation and Solve.]

In the following exercises, translate and solve.

#math.equation(block: true, alt: "The product of −6 and y is −42")[$"The product of −6 and" #h(0.2em) y #h(0.2em) "is" #h(0.2em) −42$]

#solutionbox[
−6#emph[y] = −42; #emph[y] = 7
]

#math.equation(block: true, alt: "The difference of z and −13 is −18.")[$"The difference of" #h(0.2em) z #h(0.2em) "and −13 is −18."$]

Four more than #math.equation(block: false, alt: "m")[$m$] is #math.equation(block: false, alt: "−48 .")[$−48 .$]

#solutionbox[
#emph[m] + 4 = −48; #emph[m] = −52
]

#math.equation(block: true, alt: "The product of −21 and n is 63.")[$"The product of −21 and" #h(0.2em) n #h(0.2em) "is 63."$]


==== Everyday Math

Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

#solutionbox[
Answers will vary.
]


=== Chapter Practice Test

Locate and label #math.equation(block: false, alt: "0 , 2 , −4 ,")[$0 , 2 , −4 ,$] and #math.equation(block: false, alt: "−1")[$−1$] on a number line.

In the following exercises, compare the numbers, using #math.equation(block: false, alt: "less than or greater than or =.")[$< #h(0.2em) "or" #h(0.2em) > #h(0.2em) "or" #h(0.2em) "=."$]

+ ⓐ #math.equation(block: false, alt: "−6 __ 3")[$#h(0.2em) −6 "__" 3 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "−1 __ −4")[$#h(0.2em) −1 "__" −4$]

#solutionbox[
+ ⓐ \<
+ ⓑ \>
]

+ ⓐ #math.equation(block: false, alt: "−5 __ | −5 |")[$#h(0.2em) −5 "__" | −5 | #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "minus | −2 | __ −2")[$#h(0.2em) − | −2 | "__" −2$]

In the following exercises, find the opposite of each number.

+ ⓐ #math.equation(block: false, alt: "−7")[$#h(0.2em) −7 #h(0.6em)$]
+ ⓑ #math.equation(block: false, alt: "8")[$#h(0.2em) 8$]

#solutionbox[
+ ⓐ 7
+ ⓑ −8
]

In the following exercises, simplify.

#math.equation(block: true, alt: "minus open parenthesis −22 close parenthesis")[$− ( −22 )$]

#math.equation(block: true, alt: "| 4 minus 9 |")[$| 4 − 9 |$]

#solutionbox[
5
]

#math.equation(block: true, alt: "−8 plus 6")[$−8 + 6$]

#math.equation(block: true, alt: "−15 plus open parenthesis −12 close parenthesis")[$−15 + ( −12 )$]

#solutionbox[
−27
]

#math.equation(block: true, alt: "−7 minus open parenthesis −3 close parenthesis")[$−7 − ( −3 )$]

#math.equation(block: true, alt: "10 minus open parenthesis 5 minus 6 close parenthesis")[$10 − ( 5 − 6 )$]

#solutionbox[
11
]

#math.equation(block: true, alt: "−3 times 8")[$−3 ⋅ 8$]

#math.equation(block: true, alt: "−6 open parenthesis −9 close parenthesis")[$−6 ( −9 )$]

#solutionbox[
54
]

#math.equation(block: true, alt: "70 divided by open parenthesis −7 close parenthesis")[$70 ÷ ( −7 )$]

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis cubed")[$attach(( −2 ), t: 3)$]

#solutionbox[
−8
]

#math.equation(block: true, alt: "minus 4 squared")[$− 4^(2)$]

#math.equation(block: true, alt: "16 −3 open parenthesis 5 −7 close parenthesis")[$16 −3 ( 5 −7 )$]

#solutionbox[
22
]

#math.equation(block: true, alt: "| 21 minus 6 | minus | −8 |")[$| 21 − 6 | − | −8 |$]

In the following exercises, evaluate.

#math.equation(block: true, alt: "35 minus a when a equals −4")[$35 − a #h(0.2em) "when" #h(0.2em) a = −4$]

#solutionbox[
39
]

#math.equation(block: true, alt: "open parenthesis −2 r close parenthesis squared when r equals 3")[$attach(( −2 r ), t: 2) #h(0.2em) "when" #h(0.2em) r = 3$]

#math.equation(block: true, alt: "3 m minus 2 n when m equals 6 , n equals −8")[$3 m − 2 n #h(0.2em) "when" #h(0.2em) m = 6 , #h(0.2em) n = −8$]

#solutionbox[
34
]

#math.equation(block: true, alt: "minus | minus y | when y equals 17")[$− | − y | #h(0.2em) "when" #h(0.2em) y = 17$]

In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.

the difference of −7 and −4

#solutionbox[
−7 − (−4) = −3
]

the quotient of #math.equation(block: false, alt: "25")[$25$] and the sum of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n .")[$n .$]

In the following exercises, solve.

Early one morning, the temperature in Syracuse was #math.equation(block: false, alt: "−8 °F.")[$−8 "°F."$] By noon, it had risen #math.equation(block: false, alt: "12°.")[$"12°."$] What was the temperature at noon?

#solutionbox[
4°F
]

Collette owed #math.equation(block: false, alt: "$128")[$"$128"$] on her credit card. Then she charged #math.equation(block: false, alt: "$65.")[$"$65."$] What was her new balance?

In the following exercises, solve.

#math.equation(block: true, alt: "n plus 6 equals 5")[$n + 6 = 5$]

#solutionbox[
#emph[n] = −1
]

#math.equation(block: true, alt: "p minus 11 equals −4")[$p − 11 = −4$]

#math.equation(block: true, alt: "−9 r equals −54")[$−9 r = −54$]

#solutionbox[
#emph[r] = 6
]

In the following exercises, translate and solve.

#math.equation(block: true, alt: "The product of 15 and x is 75.")[$"The product of 15 and" #h(0.2em) x #h(0.2em) "is 75."$]

#math.equation(block: true, alt: "Eight less than y is −32.")[$"Eight less than" #h(0.2em) y #h(0.2em) "is −32."$]

#solutionbox[
#emph[y] − 8 = −32; #emph[y] = −24
]