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== 3.2#h(0.6em)Add Integers

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

Evaluate #math.equation(block: false, alt: "x plus 8")[$x + 8$] when #math.equation(block: false, alt: "x equals 6 .")[$x = 6 .$] #linebreak() 
If you missed this problem, review .

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

Simplify: #math.equation(block: false, alt: "8 plus 2 open parenthesis 5 plus 1 close parenthesis .")[$8 + 2 ( 5 + 1 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "20")[$20$]
]

Translate #emph[the sum of] #math.equation(block: false, alt: "3")[$3$] #emph[and negative] #math.equation(block: false, alt: "7")[$7$] into an algebraic expression. #linebreak() 
If you missed this problem, review

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


=== Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with #strong[integers].

Most students are comfortable with the addition and subtraction facts for positive numbers.  But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of #math.equation(block: false, alt: "2 minus 1 .")[$2 − 1 .$] Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

#figure(figph[This figure has a blue circle labeled positive and a red circle labeled negative.], alt: "This figure has a blue circle labeled positive and a red circle labeled negative.", caption: none)

If we have one positive and one negative counter, the value of the pair is zero. They form a #strong[neutral pair]. The value of this neutral pair is zero as summarized in .

#figure(figph[This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.], alt: "This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.", caption: [A blue counter represents #math.equation(block: false, alt: "plus 1 .")[$+ 1 .$] A red counter represents #math.equation(block: false, alt: "−1 .")[$−1 .$] Together they add to zero.])

We will model four addition facts using the numbers #math.equation(block: false, alt: "5 , −5 and 3 , −3 .")[$5 , −5 #h(0.2em) "and" #h(0.2em) 3 #h(0.2em) , −3 .$]

#math.equation(block: true, alt: "5 plus 3 −5 plus open parenthesis −3 close parenthesis −5 plus 3 5 plus open parenthesis −3 close parenthesis")[$5 + 3 #h(1em) −5 + ( −3 ) #h(1em) −5 + 3 #h(1em) 5 + ( −3 )$]#examplebox("Example 1")[][
Model: #math.equation(block: false, alt: "5 plus 3 .")[$5 + 3 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Interpret the expression.], [#math.equation(block: false, alt: "5 plus 3")[$5 + 3$] means the sum of #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "3")[$3$].]),
  [Model the first number. Start with 5 positives.], [#figure(figph[Five light blue circles are horizontally aligned, with the number '5' positioned directly beneath the central circle, representing the total count of the circles shown.], alt: "Five light blue circles are horizontally aligned, with the number '5' positioned directly beneath the central circle, representing the total count of the circles shown.", caption: none)],
  [Model the second number. Add 3 positives.], [#figure(figph[Two groups of light blue circles are shown against a white background. The first group contains five circles, with the number '5' written below them. The second group consists of three circles, with the number '3' below.], alt: "Two groups of light blue circles are shown against a white background. The first group contains five circles, with the number '5' written below them. The second group consists of three circles, with the number '3' below.", caption: none)],
  [Count the total number of counters.], [#figure(figph[Eight light blue circles, arranged in a row, with the text '8 positives' written directly below them, illustrating a concept likely related to mathematics or data representation.], alt: "Eight light blue circles, arranged in a row, with the text '8 positives' written directly below them, illustrating a concept likely related to mathematics or data representation.", caption: none)],
  [The sum of 5 and 3 is 8.], [#math.equation(block: false, alt: "5 plus 3 equals 8")[$5 + 3 = 8$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression.

#math.equation(block: true, alt: "2 plus 4")[$2 + 4$]

#solutionbox[
#figure(figph[This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.], alt: "This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.", caption: none)

 #linebreak() 
6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression.

#math.equation(block: true, alt: "2 plus 5")[$2 + 5$]

#solutionbox[
#figure(figph[This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.], alt: "This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.", caption: none)

 #linebreak() 
7
]
]

#examplebox("Example 2")[][
Model: #math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis .")[$−5 + ( −3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Interpret the expression.], [#math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis")[$−5 + ( −3 )$] means the sum of #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "−3")[$−3$].]),
  [Model the first number. Start with 5 negatives.], [#figure(figph[Five light red circles are arranged horizontally on a white background, with the number '-5' centered beneath them.], alt: "Five light red circles are arranged horizontally on a white background, with the number '-5' centered beneath them.", caption: none)],
  [Model the second number. Add 3 negatives.], [#figure(figph[Five red circles are grouped together representing -5, while three red circles are grouped separately representing -3 on a white background.], alt: "Five red circles are grouped together representing -5, while three red circles are grouped separately representing -3 on a white background.", caption: none)],
  [Count the total number of counters.], [#figure(figph[Eight light red circles with red outlines are arranged in a row. Below them, the text '8 negatives' is written in dark blue, indicating they represent negative values.], alt: "Eight light red circles with red outlines are arranged in a row. Below them, the text '8 negatives' is written in dark blue, indicating they represent negative values.", caption: none)],
  [The sum of −5 and −3 is −8.], [#math.equation(block: false, alt: "−5 plus −3 equals −8")[$−5 + −3 = −8$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression.

#math.equation(block: true, alt: "−2 plus open parenthesis −4 close parenthesis")[$−2 + ( −4 )$]

#solutionbox[
#figure(figph[This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles.], alt: "This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles.", caption: none)

 #linebreak() −6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression.

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

#solutionbox[
#figure(figph[This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles.], alt: "This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles.", caption: none)

 #linebreak() −7
]
]

and  are very similar. The first example adds #math.equation(block: false, alt: "5")[$5$] positives and #math.equation(block: false, alt: "3")[$3$] positives—both positives. The second example adds #math.equation(block: false, alt: "5")[$5$] negatives and #math.equation(block: false, alt: "3")[$3$] negatives—both negatives. In each case, we got a result of #math.equation(block: false, alt: "8—either 8")[$"8—either" #h(0.2em) 8$] positives or #math.equation(block: false, alt: "8")[$8$] negatives. When the signs are the same, the counters are all the same color.

Now let’s see what happens when the signs are different.

#examplebox("Example 3")[][
Model: #math.equation(block: false, alt: "−5 plus 3 .")[$−5 + 3 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Interpret the expression.], [#math.equation(block: false, alt: "−5 plus 3")[$−5 + 3$] means the sum of #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "3")[$3$].]),
  [Model the first number. Start with 5 negatives.], [#figure(figph[Five light red circles, uniformly spaced, are arranged in a horizontal line against a plain white background.], alt: "Five light red circles, uniformly spaced, are arranged in a horizontal line against a plain white background.", caption: none)],
  [Model the second number. Add 3 positives.], [#figure(figph[Five red circles are arranged in a horizontal row above three light blue circles, also in a horizontal row, set against a plain white background.], alt: "Five red circles are arranged in a horizontal row above three light blue circles, also in a horizontal row, set against a plain white background.", caption: none)],
  [Remove any neutral pairs.], [#figure(figph[Three magenta ovals, each containing a red and a blue circle, are shown alongside two separate red circles.], alt: "Three magenta ovals, each containing a red and a blue circle, are shown alongside two separate red circles.", caption: none)],
  [Count the result.], [#figure(figph[A diagram displays two red-outlined pink circles, side-by-side, with the label '2 negatives' written below them on a white background, representing negative values or concepts.], alt: "A diagram displays two red-outlined pink circles, side-by-side, with the label '2 negatives' written below them on a white background, representing negative values or concepts.", caption: none)],
  [The sum of −5 and 3 is −2.], [#math.equation(block: false, alt: "−5 plus 3 equals −2")[$−5 + 3 = −2$]],
))

Notice that there were more negatives than positives, so the result is negative.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression, and then simplify:

#math.equation(block: true, alt: "2 plus open parenthesis −4 close parenthesis")[$2 + ( −4 )$]

#solutionbox[
#figure(figph[This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.], alt: "This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.", caption: none)

 #linebreak() −2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression, and then simplify:

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

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

 #linebreak() −3
]
]

#examplebox("Example 4")[][
Model: #math.equation(block: false, alt: "5 plus open parenthesis −3 close parenthesis .")[$5 + ( −3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Interpret the expression.], [#math.equation(block: false, alt: "5 plus open parenthesis −3 close parenthesis")[$5 + ( −3 )$] means the sum of #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "−3")[$−3$].]),
  [Model the first number. Start with 5 positives.], [#figure(figph[Five light blue circles with a darker blue outline are arranged in a horizontal line on a white background, resembling a star rating system or a progress indicator.], alt: "Five light blue circles with a darker blue outline are arranged in a horizontal line on a white background, resembling a star rating system or a progress indicator.", caption: none)],
  [Model the second number. Add 3 negatives.], [#figure(figph[An illustration showing five light blue circles in the top row and three red circles in the bottom row.], alt: "An illustration showing five light blue circles in the top row and three red circles in the bottom row.", caption: none)],
  [Remove any neutral pairs.], [#figure(figph[Three pairs of light blue and red circles are grouped within magenta ovals, followed by two standalone light blue circles, illustrating a mix of grouped and individual elements.], alt: "Three pairs of light blue and red circles are grouped within magenta ovals, followed by two standalone light blue circles, illustrating a mix of grouped and individual elements.", caption: none)],
  [Count the result.], [#figure(figph[Two light blue circles are displayed above the text '2 positives' on a white background, indicating a count of two positive results or items.], alt: "Two light blue circles are displayed above the text '2 positives' on a white background, indicating a count of two positive results or items.", caption: none)],
  [The sum of 5 and −3 is 2.], [#math.equation(block: false, alt: "5 plus open parenthesis −3 close parenthesis equals 2")[$5 + ( −3 ) = 2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression, and then simplify:

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis plus 4")[$( −2 ) + 4$]

#solutionbox[
#figure(figph[This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.], alt: "This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.", caption: none)

 #linebreak() 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model the expression:

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

#solutionbox[
#figure(figph[This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.], alt: "This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.", caption: none)

 #linebreak() 3
]
]

#examplebox("Example 5")[Modeling Addition of Positive and Negative Integers][
Model each addition.

+ ⓐ 4 + 2
+ ⓑ −3 + 6
+ ⓒ 4 + (−5)
+ ⓓ -2 + (−3)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "4 plus 2")[$4 + 2$]],
  [Start with 4 positives.], [#figure(figph[Four light blue circles with blue outlines are arranged horizontally on a white background, suggesting a selection or progress indicator.], alt: "Four light blue circles with blue outlines are arranged horizontally on a white background, suggesting a selection or progress indicator.", caption: none)],
  [Add two positives.], [#figure(figph[Six light blue circles with blue outlines are arranged in a horizontal row, separated into a group of four and a group of two, on a white background.], alt: "Six light blue circles with blue outlines are arranged in a horizontal row, separated into a group of four and a group of two, on a white background.", caption: none)],
  [How many do you have?], [#math.equation(block: false, alt: "4 plus 2 equals 6")[$4 + 2 = 6$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "minus 3 plus 6")[$− 3 + 6$]],
  [Start with 3 negatives.], [#figure(figph[Three light red circles with dark red outlines are arranged horizontally on a white background.], alt: "Three light red circles with dark red outlines are arranged horizontally on a white background.", caption: none)],
  [Add 6 positives.], [#figure(figph[An image displaying three red circles in the top row and six blue circles in the bottom row, representing a 3:6 ratio or comparison.], alt: "An image displaying three red circles in the top row and six blue circles in the bottom row, representing a 3:6 ratio or comparison.", caption: none)],
  [Remove neutral pairs.], [#figure(figph[An array of circles: 3 red and 3 blue circles encircled by an arrowed loop, suggesting repetition. This is followed by 3 more blue circles and an ellipsis indicating continuation.], alt: "An array of circles: 3 red and 3 blue circles encircled by an arrowed loop, suggesting repetition. This is followed by 3 more blue circles and an ellipsis indicating continuation.", caption: none)],
  [How many are left?], [#figure(figph[Three light blue circles with a blue outline, arranged horizontally on a white background.], alt: "Three light blue circles with a blue outline, arranged horizontally on a white background.", caption: none)],
  [], [#math.equation(block: false, alt: "3")[$3$]. #math.equation(block: false, alt: "−3 plus 6 equals 3")[$−3 + 6 = 3$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "4 plus open parenthesis −5 close parenthesis")[$4 + ( −5 )$]],
  [Start with 4 positives.], [#figure(figph[Four light blue circles, outlined in a slightly darker blue, are arranged horizontally on a white background.], alt: "Four light blue circles, outlined in a slightly darker blue, are arranged horizontally on a white background.", caption: none)],
  [Add 5 negatives.], [#figure(figph[Two rows of circles are displayed on a white background: four light blue circles are in the top row, and five red circles are in the bottom row.], alt: "Two rows of circles are displayed on a white background: four light blue circles are in the top row, and five red circles are in the bottom row.", caption: none)],
  [Remove neutral pairs.], [#figure(figph[Eight circles, four blue and four red, are grouped by an oval, while one red circle is separate.], alt: "Eight circles, four blue and four red, are grouped by an oval, while one red circle is separate.", caption: none)],
  [How many are left?], [#figure(figph[A single, plain light pink or pale red circle with a darker red outline is centered against a pristine white background. The simple, minimalist design highlights the single geometric shape.], alt: "A single, plain light pink or pale red circle with a darker red outline is centered against a pristine white background. The simple, minimalist design highlights the single geometric shape.", caption: none)],
  [], [#math.equation(block: false, alt: "−1")[$−1$]. #math.equation(block: false, alt: "4 plus open parenthesis −5 close parenthesis equals −1")[$4 + ( −5 ) = −1$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓓ], []),
  [], [#math.equation(block: false, alt: "−2 plus open parenthesis −3 close parenthesis")[$−2 + ( −3 )$]],
  [Start with 2 negatives.], [#figure(figph[Two light red circles with thin dark red outlines and subtle shadows are positioned horizontally close to each other on a plain white background, appearing as simple graphic elements.], alt: "Two light red circles with thin dark red outlines and subtle shadows are positioned horizontally close to each other on a plain white background, appearing as simple graphic elements.", caption: none)],
  [Add 3 negatives.], [#figure(figph[A simple graphic featuring two light red circles on the left, separated by space from three light red circles on the right, all against a white background.], alt: "A simple graphic featuring two light red circles on the left, separated by space from three light red circles on the right, all against a white background.", caption: none)],
  [How many do you have?], [#math.equation(block: false, alt: "−5")[$−5$]. #math.equation(block: false, alt: "−2 plus open parenthesis −3 close parenthesis equals −5")[$−2 + ( −3 ) = −5$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Model each addition. #linebreak()

+ ⓐ 3 + 4
+ ⓑ −1 + 4
+ ⓒ 4 + (−6)
+ ⓓ −2 + (−2)

#solutionbox[
+ ⓐ  #linebreak() #figure(figph[Two distinct groups of dark-outlined gray circles are displayed on a white background: a set of three on the left and a set of five on the right.], alt: "Two distinct groups of dark-outlined gray circles are displayed on a white background: a set of three on the left and a set of five on the right.", caption: none)
+ ⓑ   #linebreak() #figure(figph[A dark blue circle stands alone, followed by four lighter grey circles in a row, all aligned horizontally on a white background.], alt: "A dark blue circle stands alone, followed by four lighter grey circles in a row, all aligned horizontally on a white background.", caption: none)


 #linebreak()
+ ⓒ  #linebreak() #figure(figph[An abstract image featuring two rows of dark grey oval shapes, with three on the left and five on the right, against a stark white background.], alt: "An abstract image featuring two rows of dark grey oval shapes, with three on the left and five on the right, against a stark white background.", caption: none)


 #linebreak()
+ ⓓ   #linebreak() #figure(figph[Four dark gray circles, arranged in two pairs, against a white background, suggesting a simple pattern or abstract design.], alt: "Four dark gray circles, arranged in two pairs, against a white background, suggesting a simple pattern or abstract design.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ ⓐ 5 + 1
+ ⓑ −3 + 7
+ ⓒ 2 + (−8)
+ ⓓ −3 + (−4)

#solutionbox[
+ ⓐ  #linebreak() #figure(figph[Six gray circles, five connected in a row and one separate, suggesting a multi-step process or selectable items.], alt: "Six gray circles, five connected in a row and one separate, suggesting a multi-step process or selectable items.", caption: none)

 #linebreak()
+ ⓑ  #linebreak() #figure(figph[A row of dark grey and light grey oval shapes on a white background, with a gap separating the first two darker shapes from the subsequent lighter shapes.], alt: "A row of dark grey and light grey oval shapes on a white background, with a gap separating the first two darker shapes from the subsequent lighter shapes.", caption: none)

 #linebreak()
+ ⓒ  #linebreak() #figure(figph[An abstract pattern featuring a row of oval shapes. Two lighter grey ovals are on the left, separated by a white space from nine darker, blue-grey ovals that continue to the right.], alt: "An abstract pattern featuring a row of oval shapes. Two lighter grey ovals are on the left, separated by a white space from nine darker, blue-grey ovals that continue to the right.", caption: none)

 #linebreak()
+ ⓓ  #linebreak() #figure(figph[An abstract image showing eight dark grey-blue circles in two sets of four, with subtle connections, on a white background.], alt: "An abstract image showing eight dark grey-blue circles in two sets of four, with subtle connections, on a white background.", caption: none)
]
]


=== Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify #strong[expressions] with any integers.

For example, if you want to add #math.equation(block: false, alt: "37 plus open parenthesis −53 close parenthesis ,")[$37 + ( −53 ) ,$] you don’t have to count out #math.equation(block: false, alt: "37")[$37$] blue counters and #math.equation(block: false, alt: "53")[$53$] red counters.

Picture #math.equation(block: false, alt: "37")[$37$] blue counters with #math.equation(block: false, alt: "53")[$53$] red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because #math.equation(block: false, alt: "53 −37 equals 16 ,")[$53 −37 = 16 ,$] there are #math.equation(block: false, alt: "16")[$16$] more negative counters.

#math.equation(block: true, alt: "37 plus open parenthesis −53 close parenthesis equals −16")[$37 + ( −53 ) = −16$]Let’s try another one. We’ll add #math.equation(block: false, alt: "−74 plus open parenthesis −27 close parenthesis .")[$−74 + ( −27 ) .$] Imagine #math.equation(block: false, alt: "74")[$74$] red counters and #math.equation(block: false, alt: "27")[$27$] more red counters, so we have #math.equation(block: false, alt: "101")[$101$] red counters all together. This means the sum is #math.equation(block: false, alt: "−101.")[$"−101."$]

#math.equation(block: true, alt: "−74 plus open parenthesis −27 close parenthesis equals −101")[$−74 + ( −27 ) = −101$]Look again at the results of  - .

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "5 plus 3")[$5 + 3$]], [#math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis")[$−5 + ( −3 )$]],
  [both positive, sum positive], [both negative, sum negative],
  [When the signs are the same, the counters would be all the same color, so add them.], [],
  [#math.equation(block: false, alt: "−5 plus 3")[$−5 + 3$]], [#math.equation(block: false, alt: "5 plus open parenthesis −3 close parenthesis")[$5 + ( −3 )$]],
  [different signs, more negatives], [different signs, more positives],
  [Sum negative], [sum positive],
  [When the signs are different, some counters would make neutral pairs; subtract to see how many are left.], [],
))

#examplebox("Example 6")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "19 plus open parenthesis −47 close parenthesis")[$#h(0.2em) 19 + ( −47 ) #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−32 plus 40")[$#h(0.2em) −32 + 40$]

#solutionbox[
ⓐ Since the signs are different, we subtract #math.equation(block: false, alt: "19")[$19$] from #math.equation(block: false, alt: "47 .")[$47 .$] The answer will be negative because there are more negatives than positives.

#math.equation(block: true, alt: "19 plus open parenthesis −47 close parenthesis; −28")[$19 + ( −47 ) \ −28$]ⓑ The signs are different so we subtract #math.equation(block: false, alt: "32")[$32$] from #math.equation(block: false, alt: "40 .")[$40 .$] The answer will be positive because there are more positives than negatives

#math.equation(block: true, alt: "−32 plus 40; 8")[$−32 + 40 \ 8$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify each expression:

+ ⓐ #math.equation(block: false, alt: "15 plus open parenthesis −32 close parenthesis")[$#h(0.2em) 15 + ( −32 ) #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−19 plus 76")[$#h(0.2em) −19 + 76$]

#solutionbox[
+ ⓐ −17
+ ⓑ 57
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify each expression:

+ ⓐ #math.equation(block: false, alt: "−55 plus 9")[$#h(0.2em) −55 + 9 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "43 plus open parenthesis −17 close parenthesis")[$#h(0.2em) 43 + ( −17 )$]

#solutionbox[
+ ⓐ −46
+ ⓑ 26
]
]

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "−14 plus open parenthesis −36 close parenthesis .")[$−14 + ( −36 ) .$]

#solutionbox[
Since the signs are the same, we add. The answer will be negative because there are only negatives.

#math.equation(block: true, alt: "−14 plus open parenthesis −36 close parenthesis; −50")[$−14 + ( −36 ) \ −50$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expression:

#math.equation(block: true, alt: "−31 plus open parenthesis −19 close parenthesis")[$−31 + ( −19 )$]

#solutionbox[
−50
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expression:

#math.equation(block: true, alt: "−42 plus open parenthesis −28 close parenthesis")[$−42 + ( −28 )$]

#solutionbox[
−70
]
]

The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "−5 plus 3 open parenthesis −2 plus 7 close parenthesis .")[$−5 + 3 ( −2 + 7 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−5 plus 3 open parenthesis −2 plus 7 close parenthesis")[$−5 + 3 ( −2 + 7 )$]]),
  [Simplify inside the parentheses.], [#math.equation(block: false, alt: "−5 plus 3 open parenthesis 5 close parenthesis")[$−5 + 3 ( 5 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−5 plus 15")[$−5 + 15$]],
  [Add left to right.], [#math.equation(block: false, alt: "10")[$10$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expression:

#math.equation(block: true, alt: "−2 plus 5 open parenthesis −4 plus 7 close parenthesis")[$−2 + 5 ( −4 + 7 )$]

#solutionbox[
13
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expression:

#math.equation(block: true, alt: "−4 plus 2 open parenthesis −3 plus 5 close parenthesis")[$−4 + 2 ( −3 + 5 )$]

#solutionbox[
0
]
]


=== Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the #strong[variable] in the expression. Now we can use negative numbers as well as positive numbers when evaluating #strong[expressions].

#examplebox("Example 9")[][
Evaluate #math.equation(block: false, alt: "x plus 7 when")[$x + 7 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals −2")[$#h(0.2em) x = −2 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −11 .")[$#h(0.2em) #h(0.2em) x = −11 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ Evaluate #math.equation(block: false, alt: "x plus 7")[$x + 7$] when #math.equation(block: false, alt: "x equals −2")[$x = −2$]], []),
  [], [#figure(figph[A mathematical expression showing '-2 + 7'. The number -2 is in red, while the addition sign and the number 7 are in black.], alt: "A mathematical expression showing '-2 + 7'. The number -2 is in red, while the addition sign and the number 7 are in black.", caption: none)],
  [#figure(figph[The image shows the text 'Substitute -2 for x.' with '-2' highlighted in red, indicating a mathematical instruction to replace the variable x with the value -2.], alt: "The image shows the text 'Substitute -2 for x.' with '-2' highlighted in red, indicating a mathematical instruction to replace the variable x with the value -2.", caption: none)], [#figure(figph[A mathematical expression showing '-2 + 7'. The number -2 is in red, while the addition sign and the number 7 are in black.], alt: "A mathematical expression showing '-2 + 7'. The number -2 is in red, while the addition sign and the number 7 are in black.", caption: none)],
  [Simplify.], [#figure(figph[The number 5, written in black, appears in the top right corner of a plain white background.], alt: "The number 5, written in black, appears in the top right corner of a plain white background.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ Evaluate #math.equation(block: false, alt: "x plus 7")[$x + 7$] when #math.equation(block: false, alt: "x equals −11")[$x = −11$]], []),
  [], [#figure(figph[The image displays the mathematical expression 'x + 7' in a clear, sans-serif font against a white background.], alt: "The image displays the mathematical expression 'x + 7' in a clear, sans-serif font against a white background.", caption: none)],
  [#figure(figph[The image shows the mathematical instruction, 'Substitute -11 for x.' The number -11 is highlighted in red, indicating its importance in the substitution.], alt: "The image shows the mathematical instruction, 'Substitute -11 for x.' The number -11 is highlighted in red, indicating its importance in the substitution.", caption: none)], [#figure(figph[The image displays a mathematical expression showing the sum of negative eleven and positive seven, written as -11 + 7, with the negative number in red and the positive number and plus sign in black.], alt: "The image displays a mathematical expression showing the sum of negative eleven and positive seven, written as -11 + 7, with the negative number in red and the positive number and plus sign in black.", caption: none)],
  [Simplify.], [#figure(figph[A close-up view shows the number '-4' in black text against a plain white background.], alt: "A close-up view shows the number '-4' in black text against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression for the given values:

#math.equation(block: true, alt: "x plus 5 when")[$x + 5 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals −3 and")[$#h(0.2em) x = −3 #h(0.2em) "and" #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −17")[$#h(0.2em) x = −17 #h(0.2em)$]

#solutionbox[
+ ⓐ 2
+ ⓑ −12
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression for the given values: #math.equation(block: false, alt: "y plus 7")[$y + 7 #h(0.2em)$] when

+ ⓐ #math.equation(block: false, alt: "y equals −5")[$#h(0.2em) #h(0.2em) y = −5 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "y equals −8")[$#h(0.2em) y = −8 #h(0.2em)$]

#solutionbox[
+ ⓐ 2
+ ⓑ −1
]
]

#examplebox("Example 10")[][
When #math.equation(block: false, alt: "n equals −5 ,")[$n = −5 ,$] evaluate $$

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ Evaluate #math.equation(block: false, alt: "n plus 1")[$n + 1$] when #math.equation(block: false, alt: "n equals −5")[$n = −5$]], []),
  [], [#figure(figph[The mathematical expression 'n + 1' is shown in a black, bold font on a white background.], alt: "The mathematical expression 'n + 1' is shown in a black, bold font on a white background.", caption: none)],
  [#figure(figph[The text 'Substitute -5 for n.' is displayed, instructing the viewer to replace the variable 'n' with the value -5.], alt: "The text 'Substitute -5 for n.' is displayed, instructing the viewer to replace the variable 'n' with the value -5.", caption: none)], [#figure(figph[The image displays the mathematical expression '-5 + 1' with the number -5 in red and +1 in black, representing an arithmetic problem involving integers.], alt: "The image displays the mathematical expression '-5 + 1' with the number -5 in red and +1 in black, representing an arithmetic problem involving integers.", caption: none)],
  [Simplify.], [#figure(figph[The number negative four, or -4, displayed prominently on a white background.], alt: "The number negative four, or -4, displayed prominently on a white background.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ Evaluate #math.equation(block: false, alt: "minus n plus 1")[$− n + 1$] when #math.equation(block: false, alt: "n equals −5")[$n = −5$]], []),
  [], [#figure(figph[The mathematical expression -n + 1 is displayed in a clear, bold font against a white background.], alt: "The mathematical expression -n + 1 is displayed in a clear, bold font against a white background.", caption: none)],
  [#figure(figph[The text reads 'Substitute -5 for n.' with '-5' highlighted in red, indicating a specific instruction for a mathematical or variable substitution.], alt: "The text reads 'Substitute -5 for n.' with '-5' highlighted in red, indicating a specific instruction for a mathematical or variable substitution.", caption: none)], [#figure(figph[A mathematical expression shows the operation minus, followed by an opening parenthesis, then minus five, a closing parenthesis, then plus one. The number '5' and the minus sign before it are in red.], alt: "A mathematical expression shows the operation minus, followed by an opening parenthesis, then minus five, a closing parenthesis, then plus one. The number '5' and the minus sign before it are in red.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple mathematical expression '5+1' in black text against a white background, representing an addition problem.], alt: "The image displays a simple mathematical expression '5+1' in black text against a white background, representing an addition problem.", caption: none)],
  [Add.], [#figure(figph[A stark white image with a barely discernible '6' in the top right corner.], alt: "A stark white image with a barely discernible '6' in the top right corner.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
When #math.equation(block: false, alt: "n equals −8 ,")[$n = −8 ,$] evaluate

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

#solutionbox[
+ ⓐ −6
+ ⓑ 10
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "When y equals −9 , evaluate")[$"When" #h(0.2em) y = −9 , #h(0.2em) "evaluate" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "y plus 8")[$#h(0.2em) y + 8 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "minus y plus 8 .")[$#h(0.2em) − y + 8 .$]

#solutionbox[
+ ⓐ −1
+ ⓑ 17
]
]

Next we'll evaluate an expression with two variables.

#examplebox("Example 11")[][
Evaluate #math.equation(block: false, alt: "3 a plus b")[$3 a + b$] when #math.equation(block: false, alt: "a equals 12")[$a = 12$] and #math.equation(block: false, alt: "b equals −30 .")[$b = −30 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression '3a + b' is shown against a white background.], alt: "The mathematical expression '3a + b' is shown against a white background.", caption: none)]),
  [#figure(figph[The text 'Substitute 12 for a and -30 for b.' is displayed on a white background, with '12' in red and '-30' in light blue, and the rest of the text in dark gray.], alt: "The text 'Substitute 12 for a and -30 for b.' is displayed on a white background, with '12' in red and '-30' in light blue, and the rest of the text in dark gray.", caption: none)], [#figure(figph[The mathematical expression 3(12) + (-30) is displayed, with the number 12 highlighted in red and -30 highlighted in blue.], alt: "The mathematical expression 3(12) + (-30) is displayed, with the number 12 highlighted in red and -30 highlighted in blue.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the mathematical expression '36 + (-30)' in black text against a white background.], alt: "The image displays the mathematical expression '36 + (-30)' in black text against a white background.", caption: none)],
  [Add.], [#figure(figph[A plain white background featuring the number '6' subtly positioned in the top right corner.], alt: "A plain white background featuring the number '6' subtly positioned in the top right corner.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate the expression:

#math.equation(block: true, alt: "a plus 2 b when a equals −19 and b equals 14 .")[$a + 2 b #h(0.2em) "when" #h(0.2em) a = −19 #h(0.2em) "and" #h(0.2em) b = 14 .$]

#solutionbox[
9
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate the expression:

#math.equation(block: true, alt: "5 p plus q when p equals 4 and q equals −7 .")[$5 p + q #h(0.2em) "when" #h(0.2em) p = 4 #h(0.2em) "and" #h(0.2em) q = −7 .$]

#solutionbox[
13
]
]

#examplebox("Example 12")[][
Evaluate #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when #math.equation(block: false, alt: "x equals −18")[$x = −18$] and #math.equation(block: false, alt: "y equals 24 .")[$y = 24 .$]

#solutionbox[
This expression has two variables. Substitute #math.equation(block: false, alt: "−18")[$−18$] for #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "24")[$24$] for #math.equation(block: false, alt: "y .")[$y .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$]]),
  [#figure(figph[The text reads: 'Substitute -18 for x and 24 for y.' The number -18 is highlighted in red, and 24 is highlighted in light blue.], alt: "The text reads: 'Substitute -18 for x and 24 for y.' The number -18 is highlighted in red, and 24 is highlighted in light blue.", caption: none)], [#math.equation(block: false, alt: "open parenthesis −18 plus 24 close parenthesis squared")[$attach(( −18 + 24 ), t: 2)$]],
  [Add inside the parentheses.], [#math.equation(block: false, alt: "open parenthesis 6 close parenthesis squared")[$attach(( 6 ), t: 2)$]],
  [Simplify], [#math.equation(block: false, alt: "36")[$36$]],
))
]
]

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

#math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when #math.equation(block: false, alt: "x equals −15")[$x = −15$] and #math.equation(block: false, alt: "y equals 29 .")[$y = 29 .$]

#solutionbox[
196
]
]

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

#math.equation(block: false, alt: "open parenthesis x plus y close parenthesis cubed")[$attach(( x + y ), t: 3)$] when #math.equation(block: false, alt: "x equals −8")[$x = −8$] and #math.equation(block: false, alt: "y equals 10 .")[$y = 10 .$]

#solutionbox[
8
]
]


=== Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase #emph[the sum] indicates addition.

#examplebox("Example 13")[][
Translate and simplify: the sum of #math.equation(block: false, alt: "−9")[$−9$] and #math.equation(block: false, alt: "5 .")[$5 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The sum of −9 and 5 indicates addition.], [the sum of #math.equation(block: false, alt: "−9")[$−9$] and #math.equation(block: false, alt: "5")[$5$]]),
  [Translate.], [#math.equation(block: false, alt: "−9 plus 5")[$−9 + 5$]],
  [Simplify.], [#math.equation(block: false, alt: "−4")[$−4$]],
))
]
]

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

the sum of #math.equation(block: false, alt: "−7")[$−7$] and #math.equation(block: false, alt: "4")[$4$]

#solutionbox[
−7 + 4 = −3
]
]

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

the sum of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "−6")[$−6$]

#solutionbox[
−8 + (−6) = −14
]
]

#examplebox("Example 14")[][
Translate and simplify: the sum of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "−12 ,")[$−12 ,$] increased by #math.equation(block: false, alt: "3 .")[$3 .$]

#solutionbox[
The phrase #emph[increased by] indicates addition.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [The sum of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "−12")[$−12$], increased by #math.equation(block: false, alt: "3")[$3$]]),
  [Translate.], [#math.equation(block: false, alt: "[ 8 plus open parenthesis −12 close parenthesis ] plus 3")[$[ 8 + ( −12 ) ] + 3$]],
  [Simplify.], [#math.equation(block: false, alt: "−4 plus 3")[$−4 + 3$]],
  [Add.], [#math.equation(block: false, alt: "−1")[$−1$]],
))
]
]

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

the sum of #math.equation(block: false, alt: "9")[$9$] and #math.equation(block: false, alt: "−16 ,")[$−16 ,$] increased by #math.equation(block: false, alt: "4 .")[$4 .$]

#solutionbox[
\[9 + (−16)\] + 4 = −3
]
]

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

the sum of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "−12 ,")[$−12 ,$] increased by #math.equation(block: false, alt: "7 .")[$7 .$]

#solutionbox[
\[−8 + (−12)\] + 7 = −13
]
]


=== Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of #math.equation(block: false, alt: "$5")[$"$5"$] could be represented as #math.equation(block: false, alt: "−$5.")[$"−$5."$] Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

#examplebox("Example 15")[][
The temperature in Buffalo, NY, one morning started at #math.equation(block: false, alt: "7 degrees")[$7 #h(0.2em) "degrees"$] below zero Fahrenheit. By noon, it had warmed up #math.equation(block: false, alt: "12 degrees .")[$12 #h(0.2em) "degrees" .$] What was the temperature at noon?

#solutionbox[
We are asked to find the temperature at noon.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write a phrase for the temperature.], [The temperature warmed up 12 degrees from 7 degrees below zero.]),
  [Translate to math notation.], [−7 + 12],
  [Simplify.], [5],
  [Write a sentence to answer the question.], [The temperature at noon was 5 degrees Fahrenheit.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The temperature in Chicago at 5 A.M. was #math.equation(block: false, alt: "10 degrees")[$10 #h(0.2em) "degrees"$] below zero Celsius. Six hours later, it had warmed up #math.equation(block: false, alt: "14 degrees Celsius.")[$"14 degrees Celsius."$] What is the temperature at 11 A.M.?

#solutionbox[
4 degrees Celsius
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A scuba diver was swimming #math.equation(block: false, alt: "16 feet")[$"16 feet"$] below the surface and then dove down another #math.equation(block: false, alt: "17 feet.")[$"17 feet."$] What is her new depth?

#solutionbox[
−33 feet
]
]

#examplebox("Example 16")[][
A football team took possession of the football on their #math.equation(block: false, alt: "42-yard line.")[$"42-yard line."$] In the next three plays, they lost #math.equation(block: false, alt: "6 yards,")[$"6 yards,"$] gained #math.equation(block: false, alt: "4 yards,")[$"4 yards,"$] and then lost #math.equation(block: false, alt: "8 yards.")[$"8 yards."$] On what yard line was the ball at the end of those three plays?

#solutionbox[
We are asked to find the yard line the ball was on at the end of three plays.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write a word phrase for the position of the ball.], [Start at 42, then lose 6, gain 4, lose 8.]),
  [Translate to math notation.], [42 − 6 + 4 − 8],
  [Simplify.], [32],
  [Write a sentence to answer the question.], [At the end of the three plays, the ball is on the 32-yard line.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The Bears took possession of the football on their #math.equation(block: false, alt: "20-yard line.")[$"20-yard line."$] In the next three plays, they lost #math.equation(block: false, alt: "9 yards,")[$"9 yards,"$] gained #math.equation(block: false, alt: "7 yards,")[$"7 yards,"$] then lost #math.equation(block: false, alt: "4 yards.")[$"4 yards."$] On what yard line was the ball at the end of those three plays?

#solutionbox[
14-yard line
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The Chargers began with the football on their #math.equation(block: false, alt: "25-yard line.")[$"25-yard line."$] They gained #math.equation(block: false, alt: "5 yards,")[$"5 yards,"$] lost #math.equation(block: false, alt: "8 yards")[$"8 yards"$] and then gained #math.equation(block: false, alt: "15 yards")[$"15 yards"$] on the next three plays. Where was the ball at the end of these plays?

#solutionbox[
37-yard line
]
]

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

- #link("https://openstaxcollege.org/l/24samesigncount")[Adding Integers with Same Sign Using Color Counters]
- #link("https://openstaxcollege.org/l/24diffsigncount")[Adding Integers with Different Signs Using Counters]
- #link("https://openstaxcollege.org/l/24Ex1Add")[Ex1: Adding Integers]
- #link("https://openstaxcollege.org/l/24Ex2Add")[Ex2: Adding Integers]
]


=== Key Concepts

- #strong[Addition of Positive and Negative Integers]
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "5 plus 3")[$5 + 3$]], [#math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis")[$−5 + ( −3 )$]],
  [both positive, sum positive], [both negative, sum negative],
  [When the signs are the same, the counters would be all the same color, so add them.], [],
  [#math.equation(block: false, alt: "−5 plus 3")[$−5 + 3$]], [#math.equation(block: false, alt: "5 plus open parenthesis −3 close parenthesis")[$5 + ( −3 )$]],
  [different signs, more negatives], [different signs, more positives],
  [Sum negative], [sum positive],
  [When the signs are different, some counters would make neutral pairs; subtract to see how many are left.], [],
))


==== Practice Makes Perfect

#strong[Model Addition of Integers]

In the following exercises, model the expression to simplify.

#math.equation(block: true, alt: "7 plus 4")[$7 + 4$]

#solutionbox[
#figure(figph[This figure shows a row of 11 light pink circles, representing positive counters. They are separated into a group of  seven and a group of four.], alt: "This figure shows a row of 11 light pink circles, representing positive counters. They are separated into a group of seven and a group of four.", caption: none)

 #linebreak() 11
]

#math.equation(block: true, alt: "8 plus 5")[$8 + 5$]

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

#solutionbox[
#figure(figph[This figure shows a row of 9 dark pink circles, representing negative counters. They are separated into a group of six and a group of three.], alt: "This figure shows a row of 9 dark pink circles, representing negative counters. They are separated into a group of six and a group of three.", caption: none)

 #linebreak() −9
]

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

#math.equation(block: true, alt: "−7 plus 5")[$−7 + 5$]

#solutionbox[
#figure(figph[This figure shows two rows of  circles. The top row shows 7 dark pink circles, representing negative counters. The bottom row shows 5 light pink circles, representing positive counters.], alt: "This figure shows two rows of circles. The top row shows 7 dark pink circles, representing negative counters. The bottom row shows 5 light pink circles, representing positive counters.", caption: none)

 #linebreak() −2
]

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

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

#solutionbox[
#figure(figph[This figure shows two rows of circles. The top row shows 8 light pink circles, representing positive counters. The bottom row shows 7 light pink circles, representing negative counters.], alt: "This figure shows two rows of circles. The top row shows 8 light pink circles, representing positive counters. The bottom row shows 7 light pink circles, representing negative counters.", caption: none)

 #linebreak() 1
]

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

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "−21 plus open parenthesis −59 close parenthesis")[$−21 + ( −59 )$]

#solutionbox[
−80
]

#math.equation(block: true, alt: "−35 plus open parenthesis −47 close parenthesis")[$−35 + ( −47 )$]

#math.equation(block: true, alt: "48 plus open parenthesis −16 close parenthesis")[$48 + ( −16 )$]

#solutionbox[
32
]

#math.equation(block: true, alt: "34 plus open parenthesis −19 close parenthesis")[$34 + ( −19 )$]

#math.equation(block: true, alt: "−200 plus 65")[$−200 + 65$]

#solutionbox[
−135
]

#math.equation(block: true, alt: "−150 plus 45")[$−150 + 45$]

#math.equation(block: true, alt: "2 plus open parenthesis −8 close parenthesis plus 6")[$2 + ( −8 ) + 6$]

#solutionbox[
0
]

#math.equation(block: true, alt: "4 plus open parenthesis −9 close parenthesis plus 7")[$4 + ( −9 ) + 7$]

#math.equation(block: true, alt: "−14 plus open parenthesis −12 close parenthesis plus 4")[$−14 + ( −12 ) + 4$]

#solutionbox[
−22
]

#math.equation(block: true, alt: "−17 plus open parenthesis −18 close parenthesis plus 6")[$−17 + ( −18 ) + 6$]

#math.equation(block: true, alt: "135 plus open parenthesis −110 close parenthesis plus 83")[$135 + ( −110 ) + 83$]

#solutionbox[
108
]

#math.equation(block: true, alt: "140 plus open parenthesis −75 close parenthesis plus 67")[$140 + ( −75 ) + 67$]

#math.equation(block: true, alt: "−32 plus 24 plus open parenthesis −6 close parenthesis plus 10")[$−32 + 24 + ( −6 ) + 10$]

#solutionbox[
−4
]

#math.equation(block: true, alt: "−38 plus 27 plus open parenthesis −8 close parenthesis plus 12")[$−38 + 27 + ( −8 ) + 12$]

#math.equation(block: true, alt: "19 plus 2 open parenthesis −3 plus 8 close parenthesis")[$19 + 2 ( −3 + 8 )$]

#solutionbox[
29
]

#math.equation(block: true, alt: "24 plus 3 open parenthesis −5 plus 9 close parenthesis")[$24 + 3 ( −5 + 9 )$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: false, alt: "x plus 8")[$x + 8$] when

+ ⓐ #math.equation(block: false, alt: "x equals −26")[$#h(0.2em) x = −26$]
+ ⓑ #math.equation(block: false, alt: "x equals −95")[$#h(0.2em) x = −95$]

#solutionbox[
+ ⓐ −18
+ ⓑ −87
]

#math.equation(block: false, alt: "y plus 9")[$y + 9$] when

+ ⓐ #math.equation(block: false, alt: "y equals −29")[$#h(0.2em) y = −29$]
+ ⓑ #math.equation(block: false, alt: "y equals −84")[$#h(0.2em) y = −84$]

#math.equation(block: false, alt: "y plus open parenthesis −14 close parenthesis")[$y + ( −14 )$] when

+ ⓐ #math.equation(block: false, alt: "y equals −33")[$#h(0.2em) y = −33$]
+ ⓑ #math.equation(block: false, alt: "y equals 30")[$#h(0.2em) y = 30$]

#solutionbox[
+ ⓐ −47
+ ⓑ 16
]

#math.equation(block: false, alt: "x plus open parenthesis −21 close parenthesis")[$x + ( −21 )$] when

+ ⓐ #math.equation(block: false, alt: "x equals −27")[$#h(0.2em) x = −27$]
+ ⓑ #math.equation(block: false, alt: "x equals 44")[$#h(0.2em) x = 44$]

When #math.equation(block: false, alt: "a equals −7 ,")[$a = −7 ,$] evaluate:

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

#solutionbox[
+ ⓐ −4
+ ⓑ 10
]

When #math.equation(block: false, alt: "b equals −11 ,")[$b = −11 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "b plus 6")[$#h(0.2em) b + 6$]
+ ⓑ #math.equation(block: false, alt: "minus b plus 6")[$#h(0.2em) − b + 6$]

When #math.equation(block: false, alt: "c equals −9 ,")[$c = −9 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "c plus open parenthesis −4 close parenthesis")[$#h(0.2em) c + ( −4 )$]
+ ⓑ #math.equation(block: false, alt: "minus c plus open parenthesis −4 close parenthesis")[$#h(0.2em) − c + ( −4 )$]

#solutionbox[
+ ⓐ −13
+ ⓑ 5
]

When #math.equation(block: false, alt: "d equals −8 ,")[$d = −8 ,$] evaluate:

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

#math.equation(block: false, alt: "m plus n")[$m + n$] when, #math.equation(block: false, alt: "m equals −15")[$m = −15$], #math.equation(block: false, alt: "n equals 7")[$n = 7$]

#solutionbox[
−8
]

#math.equation(block: false, alt: "p plus q")[$p + q$] when, #math.equation(block: false, alt: "p equals −9")[$p = −9$], #math.equation(block: false, alt: "q equals 17")[$q = 17$]

#math.equation(block: false, alt: "r −3 s")[$r −3 s$] when, #math.equation(block: false, alt: "r equals 16")[$r = 16$], #math.equation(block: false, alt: "s equals 2")[$s = 2$]

#solutionbox[
10
]

#math.equation(block: false, alt: "2 t plus u")[$2 t + u$] when, #math.equation(block: false, alt: "t equals −6")[$t = −6$], #math.equation(block: false, alt: "u equals −5")[$u = −5$]

#math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared")[$attach(( a + b ), t: 2)$] when, #math.equation(block: false, alt: "a equals −7")[$a = −7$], #math.equation(block: false, alt: "b equals 15")[$b = 15$]

#solutionbox[
64
]

#math.equation(block: false, alt: "open parenthesis c plus d close parenthesis squared")[$attach(( c + d ), t: 2)$] when, #math.equation(block: false, alt: "c equals −5")[$c = −5$], #math.equation(block: false, alt: "d equals 14")[$d = 14$]

#math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when, #math.equation(block: false, alt: "x equals −3")[$x = −3$], #math.equation(block: false, alt: "y equals 14")[$y = 14$]

#solutionbox[
121
]

#math.equation(block: false, alt: "open parenthesis y plus z close parenthesis squared")[$attach(( y + z ), t: 2)$] when, #math.equation(block: false, alt: "y equals −3")[$y = −3$], #math.equation(block: false, alt: "z equals 15")[$z = 15$]

#strong[Translate Word Phrases to Algebraic Expressions]

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

The sum of #math.equation(block: false, alt: "−14")[$−14$] and #math.equation(block: false, alt: "5")[$5$]

#solutionbox[
−14 + 5 = −9
]

The sum of #math.equation(block: false, alt: "−22")[$−22$] and #math.equation(block: false, alt: "9")[$9$]

#math.equation(block: false, alt: "8")[$8$] more than #math.equation(block: false, alt: "−2")[$−2$]

#solutionbox[
−2 + 8 = 6
]

#math.equation(block: false, alt: "5")[$5$] more than #math.equation(block: false, alt: "−1")[$−1$]

#math.equation(block: false, alt: "−10")[$−10$] added to #math.equation(block: false, alt: "−15")[$−15$]

#solutionbox[
−15 + (−10) = −25
]

#math.equation(block: false, alt: "−6")[$−6$] added to #math.equation(block: false, alt: "−20")[$−20$]

#math.equation(block: false, alt: "6")[$6$] more than the sum of #math.equation(block: false, alt: "−1")[$−1$] and #math.equation(block: false, alt: "−12")[$−12$]

#solutionbox[
\[−1 + (−12)\] + 6 = −7
]

#math.equation(block: false, alt: "3")[$3$] more than the sum of #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "−8")[$−8$]

the sum of #math.equation(block: false, alt: "10")[$10$] and #math.equation(block: false, alt: "−19 ,")[$−19 ,$] increased by #math.equation(block: false, alt: "4")[$4$]

#solutionbox[
\[10 + (−19)\] + 4 = −5
]

the sum of #math.equation(block: false, alt: "12")[$12$] and #math.equation(block: false, alt: "−15 ,")[$−15 ,$] increased by #math.equation(block: false, alt: "1")[$1$]

#strong[Add Integers in Applications]

In the following exercises, solve.

#strong[Temperature] The temperature in St. Paul, Minnesota was #math.equation(block: false, alt: "−19 °F")[$−19 "°F"$] at sunrise. By noon the temperature had risen #math.equation(block: false, alt: "26°F.")[$"26°F."$] What was the temperature at noon?

#solutionbox[
7°F
]

#strong[Temperature] The temperature in Chicago was #math.equation(block: false, alt: "−15 °F")[$−15 "°F"$] at 6 am. By afternoon the temperature had risen #math.equation(block: false, alt: "28°F.")[$"28°F."$] What was the afternoon temperature?

#strong[Credit Cards] Lupe owes #math.equation(block: false, alt: "$73")[$"$73"$] on her credit card. Then she charges #math.equation(block: false, alt: "$45")[$"$45"$] more. What is the new balance?

#solutionbox[
−\$118
]

#strong[Credit Cards] Frank owes #math.equation(block: false, alt: "$212")[$"$212"$] on his credit card. Then he charges #math.equation(block: false, alt: "$105")[$"$105"$] more. What is the new balance?

#strong[Football] A team lost #math.equation(block: false, alt: "3 yards")[$"3 yards"$] the first play. Then they lost #math.equation(block: false, alt: "2 yards,")[$"2 yards,"$] gained #math.equation(block: false, alt: "1 yard,")[$"1 yard,"$] and then lost #math.equation(block: false, alt: "4 yards.")[$"4 yards."$] What was the change in overall yardage over the four plays?

#solutionbox[
−8 yards
]

#strong[Card Games] April lost #math.equation(block: false, alt: "5 cards")[$"5 cards"$] the first turn. Over the next three turns, she lost #math.equation(block: false, alt: "3 cards,")[$"3 cards,"$] gained #math.equation(block: false, alt: "2 cards,")[$"2 cards,"$] and then lost #math.equation(block: false, alt: "1 card.")[$"1 card."$] What was the change in cards over the four turns?

#strong[Football] The Rams took possession of the football on their own #math.equation(block: false, alt: "35-yard line.")[$"35-yard line."$] In the next three plays, they lost #math.equation(block: false, alt: "12 yards,")[$"12 yards,"$] gained #math.equation(block: false, alt: "8 yards,")[$"8 yards,"$] then lost #math.equation(block: false, alt: "6 yards.")[$"6 yards."$] On what yard line was the ball at the end of those three plays?

#solutionbox[
25-yard line
]

#strong[Football] The Cowboys began with the ball on their own #math.equation(block: false, alt: "20-yard line.")[$"20-yard line."$] They gained #math.equation(block: false, alt: "15 yards,")[$"15 yards,"$] lost #math.equation(block: false, alt: "3 yards")[$"3 yards"$] and then gained #math.equation(block: false, alt: "6 yards")[$"6 yards"$] on the next three plays. Where was the ball at the end of these plays?

#strong[Scuba Diving] A scuba diver swimming #math.equation(block: false, alt: "8 feet")[$"8 feet"$] below the surface dove #math.equation(block: false, alt: "17 feet")[$"17 feet"$] deeper; the pressure got to them and they rose five feet. What is their new depth?

#solutionbox[
20 feet
]

#strong[Gas Consumption:] Ozzie rode their motorcycle for #math.equation(block: false, alt: "30 minutes,")[$"30 minutes,"$] using #math.equation(block: false, alt: "168 fluid ounces of gas.")[$"168 fluid ounces of gas."$] Then they stopped and got #math.equation(block: false, alt: "140-fluid ounces of gas.")[$"140-fluid ounces of gas."$] Represent the change in gas amount as an integer.


==== Everyday Math

#strong[Stock Market] The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:

#math.equation(block: true, alt: "Monday, −504, Tuesday, plus 142, Wednesday, −449; Thursday, plus 410, Friday, plus 369")[$"Monday" & −504 & "Tuesday" & + 142 & "Wednesday" & −449 \ "Thursday" & + 410 & "Friday" & + 369 &$]

What was the overall change for the week?

#solutionbox[
−32
]

#strong[Stock Market] During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:

#math.equation(block: true, alt: "Monday, −201, Tuesday, −16, Wednesday, −23; Thursday, plus 172, Friday, −34")[$"Monday" & −201 & "Tuesday" & −16 & "Wednesday" & −23 \ "Thursday" & + 172 & "Friday" & −34 &$]

What was the overall change for the week?


==== Writing Exercises

Explain why the sum of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "2")[$2$] is negative, but the sum of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "−2")[$−2$] and is positive.

#solutionbox[
Sample answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive.
]

Give an example from your life experience of adding two negative numbers.


==== Self Check

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

#figure(figph[A self-assessment table for math students, allowing them to rate their confidence in skills such as modeling integer addition, simplifying expressions, evaluating variables, and translating word phrases.], alt: "A self-assessment table for math students, allowing them to rate their confidence in skills such as modeling integer addition, simplifying expressions, evaluating variables, and translating word phrases.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?