#set document(title: "3.3 Subtract Integers", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.3#h(0.6em)Subtract Integers #notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "12 minus open parenthesis 8 −1 close parenthesis .")[$12 − ( 8 −1 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "5")[$5$] ] ] #notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate #emph[the difference of] #math.equation(block: false, alt: "20")[$"20"$] #emph[and] #math.equation(block: false, alt: "−15")[$"−15"$] into an algebraic expression. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "20 minus open parenthesis minus 15 close parenthesis")[$20 − ( − 15 )$] ] ] #notebox("Be Prepared!", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: #math.equation(block: false, alt: "−18 plus 7.")[$−18 + 7 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus 11")[$− 11$] ] ] === Model Subtraction of Integers Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers. We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers. Perhaps when you were younger, you read #math.equation(block: false, alt: "5 minus 3")[$5 − 3$] as #emph[five take away three]. When we use counters, we can think of subtraction the same way. #notebox("Manipulative Mathematics", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Doing the Manipulative Mathematics activity "Subtraction of Signed Numbers" will help you develop a better understanding of subtracting integers. ] We will model four subtraction facts using the numbers #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "3.")[$3 .$] #math.equation(block: true, alt: "5 minus 3 −5 minus open parenthesis −3 close parenthesis −5 minus 35 minus 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 minus 3.")[$5 − 3 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Interpret the expression.], [#math.equation(block: false, alt: "5 minus 3")[$5 − 3$] means #math.equation(block: false, alt: "5")[$5$] take away #math.equation(block: false, alt: "3")[$3$].]), [Model the first number. Start with 5 positives.], [#figure(figph[Five light blue circles are arranged in a horizontal line, with the number '5' positioned directly below the middle circle.], alt: "Five light blue circles are arranged in a horizontal line, with the number '5' positioned directly below the middle circle.", caption: none)], [Take away the second number. So take away 3 positives.], [#figure(figph[Four light blue circles are enclosed by a pink oval and a left-pointing arrow. The text '2 positives' is written below the grouping, likely denoting a count or selection within the four.], alt: "Four light blue circles are enclosed by a pink oval and a left-pointing arrow. The text '2 positives' is written below the grouping, likely denoting a count or selection within the four.", caption: none)], [Find the counters that are left.], [#figure(figph[Two light blue circles side-by-side on a white background, resembling abstract bubbles or subtle decorative elements.], alt: "Two light blue circles side-by-side on a white background, resembling abstract bubbles or subtle decorative elements.", caption: none)], [], [#math.equation(block: false, alt: "5 minus 3 equals 2")[$5 − 3 = 2$]. #linebreak() The difference between #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "3")[$3$] is #math.equation(block: false, alt: "2")[$2$].], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "6 minus 4")[$6 − 4$] #solutionbox[ #figure(figph[This figure shows a row of 6 light pink circles, representing positive counters. The first four are circled.], alt: "This figure shows a row of 6 light pink circles, representing positive counters. The first four are circled.", caption: none) #linebreak() 2 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "7 minus 4")[$7 − 4$] #solutionbox[ #figure(figph[This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.], alt: "This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.", caption: none) #linebreak() 3 ] ] #examplebox("Example 2")[][ Model: #math.equation(block: false, alt: "−5 minus 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 minus open parenthesis −3 close parenthesis")[$−5 − ( −3 )$] means #math.equation(block: false, alt: "−5")[$−5$] take away #math.equation(block: false, alt: "−3")[$−3$].]), [Model the first number. Start with 5 negatives.], [#figure(figph[Five red circles are displayed in a row, with the number -5 written directly below the middle circles, illustrating a visual representation of the integer negative five.], alt: "Five red circles are displayed in a row, with the number -5 written directly below the middle circles, illustrating a visual representation of the integer negative five.", caption: none)], [Take away the second number. So take away 3 negatives.], [#figure(figph[A diagram shows four red circles in a row. The first three are encircled by a purple oval with an arrow. The text '2 negatives' is positioned below the fourth circle.], alt: "A diagram shows four red circles in a row. The first three are encircled by a purple oval with an arrow. The text '2 negatives' is positioned below the fourth circle.", caption: none)], [Find the number of counters that are left.], [#figure(figph[Two light red circles with darker red outlines are shown against a white background.], alt: "Two light red circles with darker red outlines are shown against a white background.", caption: none)], [], [#math.equation(block: false, alt: "−5 minus open parenthesis −3 close parenthesis equals −2")[$−5 − ( −3 ) = −2$]. #linebreak() The difference between #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "−3")[$−3$] is #math.equation(block: false, alt: "−2")[$−2$].], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "−6 minus open parenthesis −4 close parenthesis")[$−6 − ( −4 )$] #solutionbox[ #figure(figph[This figure shows a row of 6 dark pink circles, representing negative counters. The last four counters are circled.], alt: "This figure shows a row of 6 dark pink circles, representing negative counters. The last four counters are circled.", caption: none) #linebreak() −2 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "−7 minus open parenthesis −4 close parenthesis")[$−7 − ( −4 )$] #solutionbox[ #figure(figph[This figure is a row of 7 dark pink circles representing negative counters. The first four counters are circled.], alt: "This figure is a row of 7 dark pink circles representing negative counters. The first four counters are circled.", caption: none) #linebreak() −3 ] ] Notice that and are very much alike. - First, we subtracted #math.equation(block: false, alt: "3")[$3$] positives from #math.equation(block: false, alt: "5")[$5$] positives to get #math.equation(block: false, alt: "2")[$2$] positives. - Then we subtracted #math.equation(block: false, alt: "3")[$3$] negatives from #math.equation(block: false, alt: "5")[$5$] negatives to get #math.equation(block: false, alt: "2")[$2$] negatives. Each example used counters of only one color, and the “take away” model of subtraction was easy to apply. #figure(figph[This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.], alt: "This figure has a row of 5 blue circles. The first three are circled. Above the row is 5 minus 3 equals 2. Next to this is a row of 5 red circles. The first three are circled. Above the row is negative 5 minus negative 3 equals negative 2.", caption: none) Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a #strong[neutral pair] does not change the value. #examplebox("Example 3")[][ Model: #math.equation(block: false, alt: "−5 minus 3.")[$−5 − 3 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Interpret the expression.], [#math.equation(block: false, alt: "−5 minus 3")[$−5 − 3$] means #math.equation(block: false, alt: "−5")[$−5$] take away #math.equation(block: false, alt: "3")[$3$].]), [Model the first number. Start with 5 negatives.], [#figure(figph[Five light red circles with dark red outlines are arranged horizontally, with the number -5 positioned directly below the third circle from the left.], alt: "Five light red circles with dark red outlines are arranged horizontally, with the number -5 positioned directly below the third circle from the left.", caption: none)], [Take away the second number. #linebreak() So we need to take away 3 positives.], [], [But there are no positives to take away. #linebreak() Add neutral pairs until you have 3 positives.], [#figure(figph[Eight red circles and three light blue circles are shown, with five red circles on the left and a group of three red circles above three blue circles on the right.], alt: "Eight red circles and three light blue circles are shown, with five red circles on the left and a group of three red circles above three blue circles on the right.", caption: none)], [Now take away 3 positives.], [#figure(figph[An illustration of subtraction, showing three blue circles highlighted with an oval and an arrow, representing a quantity removed from a larger group of red circles.], alt: "An illustration of subtraction, showing three blue circles highlighted with an oval and an arrow, representing a quantity removed from a larger group of red circles.", caption: none)], [Count the number of counters that are left.], [#figure(figph[A diagram shows a row of eight light red circles with a darker red outline. Below the circles, the text '8 negatives' is written in a dark teal color.], alt: "A diagram shows a row of eight light red circles with a darker red outline. Below the circles, the text '8 negatives' is written in a dark teal color.", caption: none)], [], [#math.equation(block: false, alt: "−5 minus 3 equals −8")[$−5 − 3 = −8$]. #linebreak() The difference of #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "3")[$3$] is #math.equation(block: false, alt: "−8")[$−8$].], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "−6 minus 4")[$−6 − 4$] #solutionbox[ #figure(figph[This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.], alt: "This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.", caption: none) #linebreak() −10 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "−7 minus 4")[$−7 − 4$] #solutionbox[ #figure(figph[This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.], alt: "This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.", caption: none) #linebreak() −11 ] ] #examplebox("Example 4")[][ Model: #math.equation(block: false, alt: "5 minus 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 minus open parenthesis −3 close parenthesis")[$5 − ( −3 )$] means #math.equation(block: false, alt: "5")[$5$] take away #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 horizontally on a white background, resembling a progress indicator or a rating system.], alt: "Five light blue circles with a darker blue outline are arranged horizontally on a white background, resembling a progress indicator or a rating system.", caption: none)], [Take away the second number, so take away 3 negatives.], [], [But there are no negatives to take away. #linebreak() Add neutral pairs until you have 3 negatives.], [#figure(figph[Eight light blue circles and three red circles are displayed against a white background, arranged in two rows with a gap between groups.], alt: "Eight light blue circles and three red circles are displayed against a white background, arranged in two rows with a gap between groups.", caption: none)], [Then take away 3 negatives.], [#figure(figph[Three red circles are grouped by a magenta oval and arrow, signifying their removal or separation from below three of seven light blue circles, likely demonstrating a subtraction or counting concept.], alt: "Three red circles are grouped by a magenta oval and arrow, signifying their removal or separation from below three of seven light blue circles, likely demonstrating a subtraction or counting concept.", caption: none)], [Count the number of counters that are left.], [#figure(figph[Eight light blue circles with a slightly darker blue outline are arranged horizontally on a white background, with the text '8 positives' centered beneath them.], alt: "Eight light blue circles with a slightly darker blue outline are arranged horizontally on a white background, with the text '8 positives' centered beneath them.", caption: none)], [], [The difference of #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "−3")[$−3$] is #math.equation(block: false, alt: "8")[$8$]. #linebreak() #math.equation(block: false, alt: "5 minus open parenthesis −3 close parenthesis equals 8")[$5 − ( −3 ) = 8$]], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "6 minus open parenthesis −4 close parenthesis")[$6 − ( −4 )$] #solutionbox[ #figure(figph[This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.], alt: "This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.", caption: none) #linebreak() 10 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model the expression: #math.equation(block: true, alt: "7 minus open parenthesis −4 close parenthesis")[$7 − ( −4 )$] #solutionbox[ #figure(figph[This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.], alt: "This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.", caption: none) #linebreak() 11 ] ] #examplebox("Example 5")[][ Model each subtraction. #linebreak() + ⓐ 8 − 2 + ⓑ −5 − 4 + ⓒ 6 − (−6) + ⓓ −8 − (−3) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [], [#math.equation(block: false, alt: "8 minus 2")[$8 − 2$] #linebreak() This means #math.equation(block: false, alt: "8")[$8$] take away #math.equation(block: false, alt: "2")[$2$].], [Start with 8 positives.], [#figure(figph[Eight light blue circles with a subtle blue outline and shadow are arranged horizontally on a white background.], alt: "Eight light blue circles with a subtle blue outline and shadow are arranged horizontally on a white background.", caption: none)], [Take away 2 positives.], [#figure(figph[The first two light blue circles in a sequence are encircled by a magenta oval, with a curving arrow pointing back towards them, signifying a pair or a loop.], alt: "The first two light blue circles in a sequence are encircled by a magenta oval, with a curving arrow pointing back towards them, signifying a pair or a loop.", caption: none)], [How many are left?], [#math.equation(block: false, alt: "6")[$6$]], [], [#math.equation(block: false, alt: "8 minus 2 equals 6")[$8 − 2 = 6$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [], [#math.equation(block: false, alt: "−5 minus 4")[$−5 − 4$] #linebreak() This means #math.equation(block: false, alt: "−5")[$−5$] take away #math.equation(block: false, alt: "4")[$4$].], [Start with 5 negatives.], [#figure(figph[Five light red circles, each outlined in a darker red, are arranged horizontally in a row against a plain white background.], alt: "Five light red circles, each outlined in a darker red, are arranged horizontally in a row against a plain white background.", caption: none)], [You need to take away 4 positives. #linebreak() Add 4 neutral pairs to get 4 positives.], [#figure(figph[Eight light red circles with darker red borders, arranged in two groups of five and three, separated by a gap.], alt: "Eight light red circles with darker red borders, arranged in two groups of five and three, separated by a gap.", caption: none) #linebreak() #figure(figph[Four light blue circles with blue outlines, arranged horizontally on a white background, potentially indicating a loading state or steps in a process.], alt: "Four light blue circles with blue outlines, arranged horizontally on a white background, potentially indicating a loading state or steps in a process.", caption: none)], [Take away 4 positives.], [#figure(figph[Four light blue circles are grouped by a pink oval and arrow, indicating their position beneath four of the seven red circles shown above.], alt: "Four light blue circles are grouped by a pink oval and arrow, indicating their position beneath four of the seven red circles shown above.", caption: none)], [How many are left?], [#figure(figph[A series of nine light pink circles with red outlines are arranged horizontally, separated into a group of five on the left and a group of four on the right by a small gap.], alt: "A series of nine light pink circles with red outlines are arranged horizontally, separated into a group of five on the left and a group of four on the right by a small gap.", caption: none)], [], [#math.equation(block: false, alt: "−9")[$−9$]], [], [#math.equation(block: false, alt: "−5 minus 4 equals −9")[$−5 − 4 = −9$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓒ], []), [], [#math.equation(block: false, alt: "6 minus open parenthesis −6 close parenthesis")[$6 − ( −6 )$] #linebreak() This means #math.equation(block: false, alt: "6")[$6$] take away #math.equation(block: false, alt: "−6")[$−6$].], [Start with 6 positives.], [#figure(figph[Five light blue circles, outlined in a slightly darker blue, are arranged horizontally across a white background.], alt: "Five light blue circles, outlined in a slightly darker blue, are arranged horizontally across a white background.", caption: none)], [Add 6 neutrals to get 6 negatives to take away.], [#figure(figph[Two rows of circles, with 10 light blue circles on top and 6 red circles below, right-aligned, against a white background.], alt: "Two rows of circles, with 10 light blue circles on top and 6 red circles below, right-aligned, against a white background.", caption: none)], [Remove 6 negatives.], [#figure(figph[An illustration showing eleven light blue circles in a row, with five red circles directly below them. A purple oval encircles the red circles, with a curved arrow inside pointing left, suggesting a cyclical process or a moving window.], alt: "An illustration showing eleven light blue circles in a row, with five red circles directly below them. A purple oval encircles the red circles, with a curved arrow inside pointing left, suggesting a cyclical process or a moving window.", caption: none)], [How many are left?], [#figure(figph[A horizontal row of eleven light blue circles, each with a thin cyan outline, is arranged against a plain white background.], alt: "A horizontal row of eleven light blue circles, each with a thin cyan outline, is arranged against a plain white background.", caption: none)], [], [#math.equation(block: false, alt: "12")[$12$]], [], [#math.equation(block: false, alt: "6 minus open parenthesis −6 close parenthesis equals 12")[$6 − ( −6 ) = 12$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓓ], []), [], [#math.equation(block: false, alt: "−8 minus open parenthesis −3 close parenthesis")[$−8 − ( −3 )$] #linebreak() This means #math.equation(block: false, alt: "−8")[$−8$] take away #math.equation(block: false, alt: "−3")[$−3$].], [Start with 8 negatives.], [#figure(figph[Eight pale red outlined circles in a row.], alt: "Eight pale red outlined circles in a row.", caption: none)], [Take away 3 negatives.], [#figure(figph[A magenta oval groups the first three circles in a row of eight red circles, with a magenta arrow curving from the oval to the left, indicating a grouping or counting action.], alt: "A magenta oval groups the first three circles in a row of eight red circles, with a magenta arrow curving from the oval to the left, indicating a grouping or counting action.", caption: none)], [How many are left?], [#figure(figph[Five identical light red circles with a darker red outline, arranged horizontally on a plain white background.], alt: "Five identical light red circles with a darker red outline, arranged horizontally on a plain white background.", caption: none)], [], [#math.equation(block: false, alt: "−5")[$−5$]], [], [#math.equation(block: false, alt: "−8 minus open parenthesis −3 close parenthesis equals −5")[$−8 − ( −3 ) = −5$]], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model each subtraction. #linebreak() + ⓐ 7 - (-8) + ⓑ -7 - (-2) + ⓒ 4 - 1 + ⓓ -6 - 8 #solutionbox[ ⓐ #linebreak() #figure(figph[An illustration featuring two rows of circles. The bottom row shows seven dark gray circles enclosed by an oval with a left-pointing arrow, indicating a selected group or a process.], alt: "An illustration featuring two rows of circles. The bottom row shows seven dark gray circles enclosed by an oval with a left-pointing arrow, indicating a selected group or a process.", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[A row of seven grey circles, with the last two circled together and an arrow indicating a loop back to them, symbolizing repetition or a recurring cycle in a sequence.], alt: "A row of seven grey circles, with the last two circled together and an arrow indicating a loop back to them, symbolizing repetition or a recurring cycle in a sequence.", caption: none) #linebreak() ⓒ #linebreak() #figure(figph[A circular arrow highlights one grey circle in a row of four, indicating focus or a looping action on that specific element.], alt: "A circular arrow highlights one grey circle in a row of four, indicating focus or a looping action on that specific element.", caption: none) #linebreak() ⓓ #linebreak() #figure(figph[A diagram showing a row of 7 light grey circles encircled by an oval with a looping arrow, beneath two rows of 7 dark grey circles each.], alt: "A diagram showing a row of 7 light grey circles encircled by an oval with a looping arrow, beneath two rows of 7 dark grey circles each.", caption: none) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model each subtraction. #linebreak() + ⓐ 4 - (-6) + ⓑ -8 - (-1) + ⓒ 7 - 3 + ⓓ -4 - 2 #solutionbox[ ⓐ #linebreak() #figure(figph[The image shows ten light grey circles in the top row, separated into groups of four and six. Below them are six darker grey circles, which are enclosed by an oval and have a curved arrow beneath.], alt: "The image shows ten light grey circles in the top row, separated into groups of four and six. Below them are six darker grey circles, which are enclosed by an oval and have a curved arrow beneath.", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[An illustration of a finite automaton with eight states, where the final state has a self-loop, often representing an accepting or terminal state.], alt: "An illustration of a finite automaton with eight states, where the final state has a self-loop, often representing an accepting or terminal state.", caption: none) #linebreak() ⓒ #linebreak() #figure(figph[A sequence of gray circles, with the last three enclosed in an oval loop, suggesting a repeating pattern or iterative process. Four circles appear sequentially, followed by three that are part of a cycle.], alt: "A sequence of gray circles, with the last three enclosed in an oval loop, suggesting a repeating pattern or iterative process. Four circles appear sequentially, followed by three that are part of a cycle.", caption: none) #linebreak() ⓓ #linebreak() #figure(figph[An illustration featuring a row of dark grey circles, with two lighter grey circles below, encircled by an oval with a curved arrow pointing left.], alt: "An illustration featuring a row of dark grey circles, with two lighter grey circles below, encircled by an oval with a curved arrow pointing left.", caption: none) ] ] #examplebox("Example 6")[][ Model each subtraction expression: + ⓐ #math.equation(block: false, alt: "2 minus 8")[$#h(0.2em) 2 − 8$] + ⓑ #math.equation(block: false, alt: "−3 minus open parenthesis −8 close parenthesis")[$−3 − ( −8 )$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() We start with 2 positives.], [#figure(figph[Two light blue circles are displayed horizontally, with the numeral '2' printed directly beneath them, indicating a count of two.], alt: "Two light blue circles are displayed horizontally, with the numeral '2' printed directly beneath them, indicating a count of two.", caption: none)]), [We need to take away 8 positives, but we have only 2.], [], [Add neutral pairs until there are 8 positives to take away.], [#figure(figph[An array of 8 blue circles and 6 red circles, illustrating a simple numerical comparison with two blue circles separated from the main group.], alt: "An array of 8 blue circles and 6 red circles, illustrating a simple numerical comparison with two blue circles separated from the main group.", caption: none)], [Then take away eight positives.], [#figure(figph[Two rows of circles, 8 blue on top with a pink oval and left arrow, and 6 orange on the bottom, likely showing counting or a sequence.], alt: "Two rows of circles, 8 blue on top with a pink oval and left arrow, and 6 orange on the bottom, likely showing counting or a sequence.", caption: none)], [Find the number of counters that are left. #linebreak() There are 6 negatives.], [#figure(figph[Six red circles are aligned horizontally above the text '6 negatives' on a white background, likely illustrating a concept related to negative numbers or values.], alt: "Six red circles are aligned horizontally above the text '6 negatives' on a white background, likely illustrating a concept related to negative numbers or values.", caption: none)], [], [#math.equation(block: false, alt: "2 minus 8 equals −6")[$2 − 8 = −6$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ #linebreak() We start with 3 negatives.], [#figure(figph[Three red circles are arranged horizontally above the number 3, all set against a plain white background.], alt: "Three red circles are arranged horizontally above the number 3, all set against a plain white background.", caption: none)]), [We need to take away 8 negatives, but we have only 3.], [], [Add neutral pairs until there are 8 negatives to take away.], [#figure(figph[A graphic displaying red and blue circles on a white background. Eight red circles are arranged in two groups on the top row, while five blue circles form a single row below them.], alt: "A graphic displaying red and blue circles on a white background. Eight red circles are arranged in two groups on the top row, while five blue circles form a single row below them.", caption: none)], [Then take away the 8 negatives.], [#figure(figph[Eight red circles are enclosed by an oval with a leftward arrow, positioned above six blue circles, illustrating two distinct sets of objects.], alt: "Eight red circles are enclosed by an oval with a leftward arrow, positioned above six blue circles, illustrating two distinct sets of objects.", caption: none)], [Find the number of counters that are left. #linebreak() There are 5 positives.], [#figure(figph[Five blue-gray circles are arranged in a horizontal line above the text '5 positives' on a white background, suggesting a count or representation of five positive items.], alt: "Five blue-gray circles are arranged in a horizontal line above the text '5 positives' on a white background, suggesting a count or representation of five positive items.", caption: none)], [], [#math.equation(block: false, alt: "−3 minus open parenthesis −8 close parenthesis equals 5")[$−3 − ( −8 ) = 5$]], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model each subtraction expression. + ⓐ #math.equation(block: false, alt: "7 minus 9")[$#h(0.2em) 7 − 9$] + ⓑ #math.equation(block: false, alt: "−5 minus open parenthesis −9 close parenthesis")[$#h(0.2em) −5 − ( −9 )$] #solutionbox[ ⓐ #linebreak() #figure(figph[This figure shows a row of 9 light pink circles, representing positive counters. The first seven are separated from the last two. The entire row is circled. Below the last two light pink circles is a row of two dark pink circles, representing negative counters.], alt: "This figure shows a row of 9 light pink circles, representing positive counters. The first seven are separated from the last two. The entire row is circled. Below the last two light pink circles is a row of two dark pink circles, representing negative counters.", caption: none) #linebreak() −2 #linebreak() ⓑ #linebreak() #figure(figph[This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled. Below the last four dark pink circles is a row of four light pink circles, representing positive counters.], alt: "This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled. Below the last four dark pink circles is a row of four light pink circles, representing positive counters.", caption: none) #linebreak() 4 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Model each subtraction expression. + ⓐ #math.equation(block: false, alt: "4 minus 7")[$#h(0.2em) 4 − 7$] + ⓑ #math.equation(block: false, alt: "−7 minus open parenthesis −10 close parenthesis")[$#h(0.2em) −7 − ( −10 )$] #solutionbox[ ⓐ #linebreak() #figure(figph[This figure shows a row of seven light pink circles, representing positive counters. The first four are separated from the last three. The entire row is circled. Below the last three light pink circles is a row of three dark pink circles, representing negative counters.], alt: "This figure shows a row of seven light pink circles, representing positive counters. The first four are separated from the last three. The entire row is circled. Below the last three light pink circles is a row of three dark pink circles, representing negative counters.", caption: none) #linebreak() −3 #linebreak() ⓑ #linebreak() #figure(figph[This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled. Below the last three dark pink circles is a row of three light pink circles, representing positive counters.], alt: "This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled. Below the last three dark pink circles is a row of three light pink circles, representing positive counters.", caption: none) #linebreak() 3 ] ] === Simplify Expressions with Integers Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters. - Subtract #math.equation(block: false, alt: "−23 minus 7.")[$−23 − 7 .$] #linebreak() Think: We start with #math.equation(block: false, alt: "23")[$23$] negative counters. #linebreak() We have to subtract #math.equation(block: false, alt: "7")[$7$] positives, but there are no positives to take away. #linebreak() So we add #math.equation(block: false, alt: "7")[$7$] neutral pairs to get the #math.equation(block: false, alt: "7")[$7$] positives. Now we take away the #math.equation(block: false, alt: "7")[$7$] positives. #linebreak() So what’s left? We have the original #math.equation(block: false, alt: "23")[$23$] negatives plus #math.equation(block: false, alt: "7")[$7$] more negatives from the neutral pair. The result is #math.equation(block: false, alt: "30")[$30$] negatives. #linebreak() #math.equation(block: true, alt: "−23 minus 7 equals −30")[$−23 − 7 = −30$] #linebreak() Notice, that to subtract #math.equation(block: false, alt: "7,")[$"7,"$] we added #math.equation(block: false, alt: "7")[$7$] negatives. - Subtract #math.equation(block: false, alt: "30 minus open parenthesis −12 close parenthesis .")[$30 − ( −12 ) .$] #linebreak() Think: We start with #math.equation(block: false, alt: "30")[$30$] positives. #linebreak() We have to subtract #math.equation(block: false, alt: "12")[$12$] negatives, but there are no negatives to take away. #linebreak() So we add #math.equation(block: false, alt: "12")[$12$] neutral pairs to the #math.equation(block: false, alt: "30")[$30$] positives. Now we take away the #math.equation(block: false, alt: "12")[$12$] negatives. #linebreak() What’s left? We have the original #math.equation(block: false, alt: "30")[$30$] positives plus #math.equation(block: false, alt: "12")[$12$] more positives from the neutral pairs. The result is #math.equation(block: false, alt: "42")[$42$] positives. #linebreak() #math.equation(block: true, alt: "30 minus open parenthesis −12 close parenthesis equals 42")[$30 − ( −12 ) = 42$] #linebreak() Notice that to subtract #math.equation(block: false, alt: "−12 ,")[$−12 ,$] we added #math.equation(block: false, alt: "12.")[$12 .$] While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters. Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the #strong[Subtraction Property], written as follows: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Subtraction Property] #math.equation(block: true, alt: "a minus b equals a plus open parenthesis −b close parenthesis")[$a − b = a + ( "−b" )$] ] Look at these two examples. #figure(figph[This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.], alt: "This figure has two columns. The first column has 6 minus 4. Underneath, there is a row of 6 blue circles, with the first 4 separated from the last 2. The first 4 are circled. Under this row there is 2. The second column has 6 plus negative 4. Underneath there is a row of 6 blue circles with the first 4 separated from the last 2. The first 4 are circled. Under the first four is a row of 4 red circles. Under this there is 2.", caption: none) We see that #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] gives the same answer as #math.equation(block: false, alt: "6 plus open parenthesis −4 close parenthesis .")[$6 + ( −4 ) .$] Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] long ago. But knowing that #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] gives the same answer as #math.equation(block: false, alt: "6 plus open parenthesis −4 close parenthesis")[$6 + ( −4 )$] helps when we are subtracting negative numbers. #examplebox("Example 7")[][ Simplify: + ⓐ #math.equation(block: false, alt: "13 minus 8 and 13 plus open parenthesis −8 close parenthesis")[$#h(0.2em) 13 − 8 #h(0.2em) "and" #h(0.2em) 13 + ( −8 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "−17 minus 9 and −17 plus open parenthesis −9 close parenthesis")[$#h(0.2em) −17 − 9 #h(0.2em) "and" #h(0.2em) −17 + ( −9 )$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [], [#math.equation(block: false, alt: "13 minus 8")[$13 − 8$] and #math.equation(block: false, alt: "13 plus open parenthesis −8 close parenthesis")[$13 + ( −8 )$]], [Subtract to simplify.], [#math.equation(block: false, alt: "13 minus 8 equals 5")[$13 − 8 = 5$]], [Add to simplify.], [#math.equation(block: false, alt: "13 plus open parenthesis −8 close parenthesis equals 5")[$13 + ( −8 ) = 5$]], [Subtracting 8 from 13 is the same as adding −8 to 13.], [], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [], [#math.equation(block: false, alt: "−17 minus 9")[$−17 − 9$] and #math.equation(block: false, alt: "−17 plus open parenthesis −9 close parenthesis")[$−17 + ( −9 )$]], [Subtract to simplify.], [#math.equation(block: false, alt: "−17 minus 9 equals −26")[$−17 − 9 = −26$]], [Add to simplify.], [#math.equation(block: false, alt: "−17 plus open parenthesis −9 close parenthesis equals −26")[$−17 + ( −9 ) = −26$]], [Subtracting 9 from −17 is the same as adding −9 to −17.], [], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "21 minus 13 and 21 plus open parenthesis −13 close parenthesis")[$#h(0.2em) 21 − 13 #h(0.2em) "and" #h(0.2em) 21 + ( −13 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "−11 minus 7 and −11 plus open parenthesis −7 close parenthesis")[$#h(0.2em) −11 − 7 #h(0.2em) "and" #h(0.2em) −11 + ( −7 )$] #solutionbox[ + ⓐ 8, 8 + ⓑ −18, −18 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "15 minus 7 and 15 plus open parenthesis −7 close parenthesis")[$#h(0.2em) 15 − 7 #h(0.2em) "and" #h(0.2em) 15 + ( −7 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "−14 minus 8 and −14 plus open parenthesis −8 close parenthesis")[$#h(0.2em) −14 − 8 #h(0.2em) "and" #h(0.2em) −14 + ( −8 )$] #solutionbox[ + ⓐ 8, 8 + ⓑ −22, −22 ] ] Now look what happens when we subtract a negative. #figure(figph[This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.], alt: "This figure has two columns. The first column has 8 minus negative 5. Underneath, there is a row of 13 blue circles. The first 8 are separated from the next 5. Under the last 5 blue circles there is a row of 5 red circles. They are circled. Under this there is 13. The second column has 8 plus 5. Underneath there is a row of 13 blue circles. The first 8 are separated from the last 5. Under this there is 13.", caption: none) We see that #math.equation(block: false, alt: "8 minus open parenthesis −5 close parenthesis")[$8 − ( −5 )$] gives the same result as #math.equation(block: false, alt: "8 plus 5.")[$8 + 5 .$] Subtracting a negative number is like adding a positive. #examplebox("Example 8")[][ Simplify: + ⓐ #math.equation(block: false, alt: "9 minus open parenthesis −15 close parenthesis and 9 plus 15")[$#h(0.2em) 9 − ( −15 ) #h(0.2em) "and" #h(0.2em) 9 + 15 #h(1em)$] + ⓑ #math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis and −7 plus 4")[$#h(0.2em) −7 − ( −4 ) #h(0.2em) "and" #h(0.2em) −7 + 4$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [], [#math.equation(block: false, alt: "9 minus open parenthesis −15 close parenthesis")[$9 − ( −15 )$] and #math.equation(block: false, alt: "9 plus 15")[$9 + 15$]], [Subtract to simplify.], [#math.equation(block: false, alt: "9 minus open parenthesis −15 close parenthesis equals 24")[$9 − ( −15 ) = 24$]], [Add to simplify.], [#math.equation(block: false, alt: "9 plus 15 equals 24")[$9 + 15 = 24$]], [Subtracting −15 from 9 is the same as adding 15 to 9.], [], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [], [#math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis")[$−7 − ( −4 )$] and #math.equation(block: false, alt: "−7 plus 4")[$−7 + 4$]], [Subtract to simplify.], [#math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis equals −3")[$−7 − ( −4 ) = −3$]], [Add to simplify.], [#math.equation(block: false, alt: "−7 plus 4 equals −3")[$−7 + 4 = −3$]], [Subtracting −4 from −7 is the same as adding 4 to −7], [], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "6 minus open parenthesis −13 close parenthesis and 6 plus 13")[$#h(0.2em) 6 − ( −13 ) #h(0.2em) "and" #h(0.2em) 6 + 13 #h(1em)$] + ⓑ #math.equation(block: false, alt: "−5 minus open parenthesis −1 close parenthesis and −5 plus 1")[$#h(0.2em) −5 − ( −1 ) #h(0.2em) "and" #h(0.2em) −5 + 1$] #solutionbox[ + ⓐ 19, 19 + ⓑ −4, −4 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify each expression: + ⓐ #math.equation(block: false, alt: "4 minus open parenthesis −19 close parenthesis and 4 plus 19")[$#h(0.2em) 4 − ( −19 ) #h(0.2em) "and" #h(0.2em) 4 + 19 #h(1em)$] + ⓐ #math.equation(block: false, alt: "−4 minus open parenthesis −7 close parenthesis and −4 plus 7")[$−4 − ( −7 ) #h(0.2em) "and" #h(0.2em) −4 + 7$] #solutionbox[ + ⓐ 23, 23 + ⓑ 3, 3 ] ] Look again at the results of - . #figure(table( columns: 2, align: left, inset: 6pt, [#math.equation(block: false, alt: "5 minus 3")[$5 − 3$]], [#math.equation(block: false, alt: "–5 minus open parenthesis –3 close parenthesis")[$–5 − ( –3 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "–2")[$–2$]], [2 positives], [2 negatives], [When there would be enough counters of the color to take away, subtract.], [], [#math.equation(block: false, alt: "–5 minus 3")[$–5 − 3$]], [#math.equation(block: false, alt: "5 minus open parenthesis –3 close parenthesis")[$5 − ( –3 )$]], [#math.equation(block: false, alt: "–8")[$–8$]], [#math.equation(block: false, alt: "8")[$8$]], [5 negatives, want to subtract 3 positives], [5 positives, want to subtract 3 negatives], [need neutral pairs], [need neutral pairs], [When there would not be enough of the counters to take away, add neutral pairs.], [], )) #examplebox("Example 9")[][ Simplify: #math.equation(block: false, alt: "−74 minus open parenthesis −58 close parenthesis .")[$−74 − ( −58 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([We are taking 58 negatives away from 74 negatives.], [#math.equation(block: false, alt: "−74 minus open parenthesis −58 close parenthesis")[$−74 − ( −58 )$]]), [Subtract.], [#math.equation(block: false, alt: "−16")[$−16$]], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "−67 minus open parenthesis −38 close parenthesis")[$−67 − ( −38 )$] #solutionbox[ −29 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "−83 minus open parenthesis −57 close parenthesis")[$−83 − ( −57 )$] #solutionbox[ −26 ] ] #examplebox("Example 10")[][ Simplify: #math.equation(block: false, alt: "7 minus open parenthesis −4 minus 3 close parenthesis minus 9.")[$7 − ( −4 − 3 ) − 9 .$] #solutionbox[ We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Simplify inside the parentheses first.], [#figure(figph[A mathematical expression reads '7 minus open parenthesis minus 4 minus 3 close parenthesis minus 9.'], alt: "A mathematical expression reads '7 minus open parenthesis minus 4 minus 3 close parenthesis minus 9.'", caption: none)]), [Subtract from left to right.], [#figure(figph[A mathematical expression showing 7 minus negative 7 minus 9.], alt: "A mathematical expression showing 7 minus negative 7 minus 9.", caption: none)], [Subtract.], [#figure(figph[The mathematical expression '14 - 9' is displayed in a clear, dark font against a plain white background, ready for calculation.], alt: "The mathematical expression '14 - 9' is displayed in a clear, dark font against a plain white background, ready for calculation.", caption: none)], [], [#figure(figph[The black numeral '5' is prominently displayed against a plain white background.], alt: "The black numeral '5' is prominently displayed against a plain white background.", caption: none)], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "8 minus open parenthesis −3 minus 1 close parenthesis minus 9")[$8 − ( −3 − 1 ) − 9$] #solutionbox[ 3 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "12 minus open parenthesis −9 minus 6 close parenthesis minus 14")[$12 − ( −9 − 6 ) − 14$] #solutionbox[ 13 ] ] #examplebox("Example 11")[][ Simplify: #math.equation(block: false, alt: "3 times 7 minus 4 times 7 minus 5 times 8.")[$3 · 7 − 4 · 7 − 5 · 8 .$] #solutionbox[ We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Multiply first.], [#figure(figph[The image shows the mathematical expression '3.7 - 4.7 - 5.8' in a horizontal line, rendered in a plain, clear font against a white background.], alt: "The image shows the mathematical expression '3.7 - 4.7 - 5.8' in a horizontal line, rendered in a plain, clear font against a white background.", caption: none)]), [Subtract from left to right.], [#figure(figph[The numbers 21, 28, and 40 are displayed in a sequence, separated by hyphens, against a white background.], alt: "The numbers 21, 28, and 40 are displayed in a sequence, separated by hyphens, against a white background.", caption: none)], [Subtract.], [#figure(figph[The image shows the mathematical expression -7 - 40, centered against a white background.], alt: "The image shows the mathematical expression -7 - 40, centered against a white background.", caption: none)], [], [#figure(figph[The number -47 is displayed in black text on a white background.], alt: "The number -47 is displayed in black text on a white background.", caption: none)], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "6 times 2 minus 9 times 1 minus 8 times 9.")[$6 · 2 − 9 · 1 − 8 · 9 .$] #solutionbox[ −69 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify the expression: #math.equation(block: true, alt: "2 times 5 minus 3 times 7 minus 4 times 9")[$2 · 5 − 3 · 7 − 4 · 9$] #solutionbox[ −47 ] ] === Evaluate Variable Expressions with Integers Now we’ll practice evaluating #strong[expressions] that involve #strong[subtracting negative numbers] as well as positive numbers. #examplebox("Example 12")[][ Evaluate #math.equation(block: false, alt: "x minus 4 when")[$x − 4 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "x equals −6.")[$#h(0.2em) x = −6 .$] #solutionbox[ ⓐ To evaluate #math.equation(block: false, alt: "x minus 4")[$x − 4$] when #math.equation(block: false, alt: "x equals 3")[$x = 3$], substitute #math.equation(block: false, alt: "3")[$3$] for #math.equation(block: false, alt: "x")[$x$] in the expression. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical expression 'x - 4' in black text against a white background. The variable 'x' is followed by a minus sign, and then the numeral '4' completes the simple algebraic expression.], alt: "The image displays the mathematical expression 'x - 4' in black text against a white background. The variable 'x' is followed by a minus sign, and then the numeral '4' completes the simple algebraic expression.", caption: none)]), [#figure(figph[The image displays the text 'Substitute 3 for x.' in a bold, sans-serif font against a white background. The number '3' is highlighted in red, while the rest of the text is a dark, desaturated teal color.], alt: "The image displays the text 'Substitute 3 for x.' in a bold, sans-serif font against a white background. The number '3' is highlighted in red, while the rest of the text is a dark, desaturated teal color.", caption: none)], [#figure(figph[The mathematical expression '3 - 4' is displayed, showing a subtraction operation between the numbers three and four.], alt: "The mathematical expression '3 - 4' is displayed, showing a subtraction operation between the numbers three and four.", caption: none)], [Subtract.], [#figure(figph[The image displays the number '-1' in black font against a white background.], alt: "The image displays the number '-1' in black font against a white background.", caption: none)], )) ⓑ To evaluate #math.equation(block: false, alt: "x minus 4")[$x − 4$] when #math.equation(block: false, alt: "x equals −6 ,")[$x = −6 ,$] substitute #math.equation(block: false, alt: "−6")[$−6$] for #math.equation(block: false, alt: "x")[$x$] in the expression. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression 'x - 4' is displayed in black font against a white background.], alt: "The mathematical expression 'x - 4' is displayed in black font against a white background.", caption: none)]), [#figure(figph[The text 'Substitute -6 for x.' is displayed in a blue-green gradient font on a white background, with the '-6' highlighted in red.], alt: "The text 'Substitute -6 for x.' is displayed in a blue-green gradient font on a white background, with the '-6' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression displays '-6 - 4' in red and black text on a white background, representing the subtraction of 4 from -6.], alt: "A mathematical expression displays '-6 - 4' in red and black text on a white background, representing the subtraction of 4 from -6.", caption: none)], [Subtract.], [#figure(figph[The number negative 10, or minus 10, is displayed in black text on a white background.], alt: "The number negative 10, or minus 10, is displayed in black text on a white background.", caption: none)], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate each expression: #math.equation(block: true, alt: "y minus 7 when")[$y − 7 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "y equals 5")[$#h(0.2em) y = 5 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "y equals −8")[$#h(0.2em) y = −8$] #solutionbox[ + ⓐ −2 + ⓑ −15 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate each expression: #math.equation(block: true, alt: "m minus 3 when")[$m − 3 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "m equals 1")[$#h(0.2em) m = 1 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "m equals −4")[$#h(0.2em) m = −4$] #solutionbox[ + ⓐ −2 + ⓑ −7 ] ] #examplebox("Example 13")[][ Evaluate #math.equation(block: false, alt: "20 minus z when")[$20 − z #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "z equals 12")[$#h(0.2em) z = 12 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "z equals minus 12")[$#h(0.2em) z = − 12$] #solutionbox[ ⓐ To evaluate #math.equation(block: false, alt: "20 minus z when z equals 12 ,")[$20 − z #h(0.2em) "when" #h(0.2em) z = 12 ,$] substitute #math.equation(block: false, alt: "12")[$12$] for #math.equation(block: false, alt: "z")[$z$] in the expression. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displays '20 - z' in a simple, clear font against a white background.], alt: "A mathematical expression displays '20 - z' in a simple, clear font against a white background.", caption: none)]), [#figure(figph[The image shows the text 'Substitute 12 for z.' The word 'Substitute' is dark blue-green, the number '12' is red, and 'for z.' is dark blue-green. The text suggests a mathematical or algebraic instruction.], alt: "The image shows the text 'Substitute 12 for z.' The word 'Substitute' is dark blue-green, the number '12' is red, and 'for z.' is dark blue-green. The text suggests a mathematical or algebraic instruction.", caption: none)], [#figure(figph[A mathematical expression showing '20 - 12', with the number '12' highlighted in red.], alt: "A mathematical expression showing '20 - 12', with the number '12' highlighted in red.", caption: none)], [Subtract.], [#figure(figph[A single black numeral '8' is displayed against a stark white background.], alt: "A single black numeral '8' is displayed against a stark white background.", caption: none)], )) ⓑ To evaluate #math.equation(block: false, alt: "20 minus z when z equals −12 , substitute −12 for z in the expression.")[$20 − z #h(0.2em) "when" #h(0.2em) z = −12 , #h(0.2em) "substitute" #h(0.2em) −12 #h(0.2em) "for" #h(0.2em) z #h(0.2em) "in the expression."$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the algebraic expression '20 - z' in a simple, clear font against a white background.], alt: "The image displays the algebraic expression '20 - z' in a simple, clear font against a white background.", caption: none)]), [#figure(figph[The text reads 'Substitute -12 for z.', with '-12' highlighted in red.], alt: "The text reads 'Substitute -12 for z.', with '-12' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression displays '20 - (-12)' in black and red text against a white background.], alt: "A mathematical expression displays '20 - (-12)' in black and red text against a white background.", caption: none)], [Subtract.], [#figure(figph[The number 32 is displayed prominently in a gray font on a plain white background, centered within the image frame. The digits appear to be a standard, sans-serif typeface, clean and legible.], alt: "The number 32 is displayed prominently in a gray font on a plain white background, centered within the image frame. The digits appear to be a standard, sans-serif typeface, clean and legible.", caption: none)], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate each expression: #math.equation(block: true, alt: "17 minus k when")[$17 − k #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "k equals 19")[$#h(0.2em) k = 19 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "k equals −19")[$#h(0.2em) k = −19$] #solutionbox[ + ⓐ −2 + ⓑ 36 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate each expression: #math.equation(block: true, alt: "−5 minus b when")[$−5 − b #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "b equals 14")[$#h(0.2em) b = 14 #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "b equals −14")[$#h(0.2em) b = −14$] #solutionbox[ + ⓐ −19 + ⓑ 9 ] ] === Translate Word Phrases to Algebraic Expressions When we first introduced the operation symbols, we saw that the expression #math.equation(block: false, alt: "a minus b")[$a − b$] may be read in several ways as shown below. #figure(figph[This table has six rows. The first row has a - b. The second row states a minus b. The third row states the difference of a and b. The fourth row states subtract b from a. The fifth row states b subtracted from a. The sixth row states b less than a.], alt: "This table has six rows. The first row has a - b. The second row states a minus b. The third row states the difference of a and b. The fourth row states subtract b from a. The fifth row states b subtracted from a. The sixth row states b less than a.", caption: none) Be careful to get #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] in the right order! #examplebox("Example 14")[][ Translate and then simplify: + ⓐ the difference of #math.equation(block: false, alt: "13")[$13$] and #math.equation(block: false, alt: "−21")[$−21$] + ⓑ subtract #math.equation(block: false, alt: "24")[$24$] from #math.equation(block: false, alt: "−19")[$−19$] #solutionbox[ ⓐ A #emph[difference] means subtraction. Subtract the numbers in the order they are given. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The text 'the difference of 13 and -21' is displayed, with the word 'difference' highlighted in red.], alt: "The text 'the difference of 13 and -21' is displayed, with the word 'difference' highlighted in red.", caption: none)]), [Translate.], [#figure(figph[A mathematical expression '13 - (-21)' is displayed on a white background, demonstrating the subtraction of a negative number.], alt: "A mathematical expression '13 - (-21)' is displayed on a white background, demonstrating the subtraction of a negative number.", caption: none)], [Simplify.], [#figure(figph[The number 34 is prominently displayed in the upper right portion of a white background. The digits are in a dark color, providing a clear contrast against the bright, plain surface.], alt: "The number 34 is prominently displayed in the upper right portion of a white background. The digits are in a dark color, providing a clear contrast against the bright, plain surface.", caption: none)], )) ⓑ #emph[Subtract] means to take #math.equation(block: false, alt: "24")[$24$] away from #math.equation(block: false, alt: "−19.")[$−19 .$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The text 'subtract 24 from -19' is displayed, instructing a mathematical operation.], alt: "The text 'subtract 24 from -19' is displayed, instructing a mathematical operation.", caption: none)]), [Translate.], [#figure(figph[The image displays a mathematical expression '-19 - 24' in a dark gray font on a plain white background, presenting a simple arithmetic problem.], alt: "The image displays a mathematical expression '-19 - 24' in a dark gray font on a plain white background, presenting a simple arithmetic problem.", caption: none)], [Simplify.], [#figure(figph[The number -43 is displayed in black text on a plain white background.], alt: "The number -43 is displayed in black text on a plain white background.", caption: none)], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and simplify: + ⓐ the difference of #math.equation(block: false, alt: "14")[$14$] and #math.equation(block: false, alt: "−23")[$−23$] + ⓑ subtract #math.equation(block: false, alt: "21")[$21$] from #math.equation(block: false, alt: "−17")[$−17$] #solutionbox[ + ⓐ 14 − (−23) = 37 + ⓑ −17 − 21 = −38 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate and simplify: + ⓐ the difference of #math.equation(block: false, alt: "11")[$11$] and #math.equation(block: false, alt: "−19")[$−19$] + ⓑ subtract #math.equation(block: false, alt: "18")[$18$] from #math.equation(block: false, alt: "−11")[$−11$] #solutionbox[ + ⓐ 11 − (−19) = 30 + ⓑ −11 − 18 = −29 ] ] === Subtract Integers in Applications It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense. #notebox("How To", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve Application Problems.] + Identify what you are asked to find. + Write a phrase that gives the information to find it. + Translate the phrase to an expression. + Simplify the expression. + Answer the question with a complete sentence. ] #examplebox("Example 15")[][ In the morning, the temperature in Urbana, Illinois was #math.equation(block: false, alt: "11")[$11$] degrees Fahrenheit. By mid-afternoon, the temperature had dropped to #math.equation(block: false, alt: "−9")[$−9$] degrees Fahrenheit. What was the difference between the morning and afternoon temperatures? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. Identify what we are asked to find.], [the difference between the morning and afternoon temperatures]), [Step 2. Write a phrase that gives the information to find it.], [the difference of #math.equation(block: false, alt: "11")[$11$] and #math.equation(block: false, alt: "−9")[$−9$]], [Step 3. Translate the phrase to an expression. #linebreak() The word #emph[difference] indicates subtraction.], [#math.equation(block: false, alt: "11 minus open parenthesis −9 close parenthesis")[$11 − ( −9 )$]], [Step 4. Simplify the expression.], [#math.equation(block: false, alt: "20")[$20$]], [Step 5. Write a complete sentence that answers the question.], [The difference in temperature was #math.equation(block: false, alt: "20")[$20$] degrees Fahrenheit.], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ In the morning, the temperature in Anchorage, Alaska was #math.equation(block: false, alt: "15 degrees Fahrenheit.")[$"15 degrees Fahrenheit."$] By mid-afternoon the temperature had dropped to #math.equation(block: false, alt: "30 degrees")[$30 "degrees"$] below zero. What was the difference between the morning and afternoon temperatures? #solutionbox[ 45 degrees Fahrenheit ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The temperature in Denver was #math.equation(block: false, alt: "−6")[$−6$] degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to #math.equation(block: false, alt: "−15 degree Fahrenheit.")[$"−15 degree Fahrenheit."$] What was the difference between the lunchtime and sunset temperatures? #solutionbox[ 9 degrees Fahrenheit ] ] Geography provides another application of negative numbers with the elevations of places below sea level. #examplebox("Example 16")[][ Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is #math.equation(block: false, alt: "14,497")[$14,497$] feet above sea level and the elevation of Death Valley is #math.equation(block: false, alt: "282")[$282$] feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. What are we asked to find?], [The difference in elevation between Mt. Whitney and Death Valley]), [Step 2. Write a phrase.], [elevation of Mt. Whitney−elevation of Death Valley], [Step 3. Translate.], [#math.equation(block: false, alt: "14,497 minus open parenthesis −282 close parenthesis")[$14,497 − ( −282 )$]], [Step 4. Simplify.], [#math.equation(block: false, alt: "14,779")[$14,779$]], [Step 5. Write a complete sentence that answers the question.], [The difference in elevation is #math.equation(block: false, alt: "14,779")[$14,779$] feet.], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ One day, John hiked to the #math.equation(block: false, alt: "10,023 foot")[$10,023 "foot"$] summit of Haleakala volcano in Hawaii. The next day, while scuba diving, he dove to a cave #math.equation(block: false, alt: "80 feet")[$80 "feet"$] below sea level. What is the difference between the elevation of the summit of Haleakala and the depth of the cave? #solutionbox[ 10,103 feet ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The submarine Nautilus is at #math.equation(block: false, alt: "340 feet")[$340 "feet"$] below the surface of the water and the submarine Explorer is #math.equation(block: false, alt: "573 feet")[$573 "feet"$] below the surface of the water. What is the difference in the position of the Nautilus and the Explorer? #solutionbox[ 233 feet ] ] Managing your money can involve both positive and negative numbers. You might have #strong[overdraft] protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!) #examplebox("Example 17")[][ Leslie has #math.equation(block: false, alt: "$25")[$"$25"$] in her checking account and she writes a check for #math.equation(block: false, alt: "$8.")[$"$8."$] + ⓐ What is the balance after she writes the check? + ⓑ She writes a second check for #math.equation(block: false, alt: "$20.")[$"$20."$] What is the new balance after this check? + ⓒ Leslie’s friend told her that she had lost a check for #math.equation(block: false, alt: "$10")[$"$10"$] that Leslie had given her with her birthday card. What is the balance in Leslie’s checking account now? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [What are we asked to find?], [The balance of the account], [Write a phrase.], [#math.equation(block: false, alt: "$25")[$\$25$] minus #math.equation(block: false, alt: "$8")[$\$8$]], [Translate], [#figure(figph[The image displays a mathematical subtraction problem, written as '\$25 - \$8', indicating an operation to find the difference between twenty-five dollars and eight dollars.], alt: "The image displays a mathematical subtraction problem, written as '$25 - $8', indicating an operation to find the difference between twenty-five dollars and eight dollars.", caption: none)], [Simplify.], [#figure(figph[The number '\$17' is displayed in black text on a plain white background, indicating a monetary value of seventeen dollars.], alt: "The number '$17' is displayed in black text on a plain white background, indicating a monetary value of seventeen dollars.", caption: none)], [Write a sentence answer.], [The balance is #math.equation(block: false, alt: "$ 17")[$\$ 17$].], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [What are we asked to find?], [The new balance], [Write a phrase.], [#math.equation(block: false, alt: "$17")[$\$17$] minus #math.equation(block: false, alt: "$20")[$\$20$]], [Translate], [#figure(figph[A numerical text image displays a price range of '\$17 - \$20' in black font on a white background.], alt: "A numerical text image displays a price range of '$17 - $20' in black font on a white background.", caption: none)], [Simplify.], [#figure(figph[The image displays '-\$3' in black text against a plain white background, indicating a negative financial value or a cost of three dollars.], alt: "The image displays '-$3' in black text against a plain white background, indicating a negative financial value or a cost of three dollars.", caption: none)], [Write a sentence answer.], [She is overdrawn by #math.equation(block: false, alt: "$ 3")[$\$ 3$].], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓒ], []), [What are we asked to find?], [The new balance], [Write a phrase.], [#math.equation(block: false, alt: "$10")[$\$10$] more than #math.equation(block: false, alt: "minus $3")[$− \$3$]], [Translate], [#figure(figph[The image displays a mathematical expression showing a sum: -\$3 + \$10. This represents adding a positive \$10 to a negative \$3, resulting in a net positive value of \$7.], alt: "The image displays a mathematical expression showing a sum: -$3 + $10. This represents adding a positive $10 to a negative $3, resulting in a net positive value of $7.", caption: none)], [Simplify.], [#figure(figph[The image shows the text \$7, indicating a price or monetary value of seven dollars.], alt: "The image shows the text $7, indicating a price or monetary value of seven dollars.", caption: none)], [Write a sentence answer.], [The balance is now #math.equation(block: false, alt: "$ 7")[$\$ 7$].], )) ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Araceli has #math.equation(block: false, alt: "$75")[$"$75"$] in her checking account and writes a check for #math.equation(block: false, alt: "$27.")[$"$27" .$] + ⓐ What is the balance after she writes the check? + ⓑ She writes a second check for #math.equation(block: false, alt: "$50.")[$"$50" .$] What is the new balance? + ⓒ The check for #math.equation(block: false, alt: "$20")[$"$20"$] that she sent a charity was never cashed. What is the balance in Araceli’s checking account now? #solutionbox[ + ⓐ \$48 + ⓑ −\$2 + ⓒ \$18 ] ] #notebox("Try It", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Genevieve’s bank account was overdrawn and the balance is #math.equation(block: false, alt: "minus $78.")[$− "$78" .$] + ⓐ She deposits a check for #math.equation(block: false, alt: "$24")[$"$24"$] that she earned babysitting. What is the new balance? + ⓑ She deposits another check for #math.equation(block: false, alt: "$49.")[$"$49" .$] Is she out of debt yet? What is her new balance? #solutionbox[ + ⓐ −\$54 + ⓑ No, −\$5 ] ] #notebox("Links to Literacy", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The Links to Literacy activity "Elevator Magic" will provide you with another view of the topics covered in this section. ] #notebox("Media", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://openstaxcollege.org/l/24AddSubtrInteg")[Adding and Subtracting Integers] - #link("https://openstaxcollege.org/l/24Subtrinteger")[Subtracting Integers with Color Counters] - #link("https://openstaxcollege.org/l/24Subtractbasic")[Subtracting Integers Basics] - #link("https://openstaxcollege.org/l/24introintegerr")[Subtracting Integers] - #link("https://openstaxcollege.org/l/24integerappp")[Integer Application] ] === Key Concepts - #strong[Subtraction of Integers]#figure(table( columns: 2, align: left, inset: 6pt, [#math.equation(block: false, alt: "5 minus 3")[$5 − 3$]], [#math.equation(block: false, alt: "–5 minus open parenthesis –3 close parenthesis")[$–5 − ( –3 )$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "–2")[$–2$]], [2 positives], [2 negatives], [When there would be enough counters of the color to take away, subtract.], [], [#math.equation(block: false, alt: "–5 minus 3")[$–5 − 3$]], [#math.equation(block: false, alt: "5 minus open parenthesis –3 close parenthesis")[$5 − ( –3 )$]], [#math.equation(block: false, alt: "–8")[$–8$]], [#math.equation(block: false, alt: "8")[$8$]], [5 negatives, want to subtract 3 positives], [5 positives, want to subtract 3 negatives], [need neutral pairs], [need neutral pairs], [When there would not be enough of the counters to take away, add neutral pairs.], [], )) - #strong[Subtraction Property] - #math.equation(block: false, alt: "a minus b equals a plus open parenthesis −b close parenthesis")[$a − b = a + ( "−b" )$] - #math.equation(block: false, alt: "a minus open parenthesis −b close parenthesis equals a plus b")[$a − ( "−b" ) = a + b$] - #strong[Solve Application Problems] - Step 1. Identify what you are asked to find. - Step 2. Write a phrase that gives the information to find it. - Step 3. Translate the phrase to an expression. - Step 4. Simplify the expression. - Step 5. Answer the question with a complete sentence. === Practice Makes Perfect #strong[Model Subtraction of Integers] In the following exercises, model each expression and simplify. #math.equation(block: true, alt: "8 minus 2")[$8 − 2$] #solutionbox[ #figure(figph[This figure shows a row of 8 light pink circles, representing positive counters. The first 2 are circles and are separated from the last 6.], alt: "This figure shows a row of 8 light pink circles, representing positive counters. The first 2 are circles and are separated from the last 6.", caption: none) #linebreak() 6 ] #math.equation(block: true, alt: "9 minus 3")[$9 − 3$] #math.equation(block: true, alt: "−5 minus open parenthesis −1 close parenthesis")[$−5 − ( −1 )$] #solutionbox[ #figure(figph[This figure ishows a row of 5 dark pink circles. The first one is circled.], alt: "This figure ishows a row of 5 dark pink circles. The first one is circled.", caption: none) #linebreak() −4 ] #math.equation(block: true, alt: "−6 minus open parenthesis −4 close parenthesis")[$−6 − ( −4 )$] #math.equation(block: true, alt: "−5 minus 4")[$−5 − 4$] #solutionbox[ #figure(figph[This figure has a row of 9 dark pink circles representing negative counters. The first 5 are separated from the last 4. Below the last 4 is a row of 4 light pink circles, representing positive counters. These four positive counters are circled.], alt: "This figure has a row of 9 dark pink circles representing negative counters. The first 5 are separated from the last 4. Below the last 4 is a row of 4 light pink circles, representing positive counters. These four positive counters are circled.", caption: none) #linebreak() −9 ] #math.equation(block: true, alt: "−7 minus 2")[$−7 − 2$] #math.equation(block: true, alt: "8 minus open parenthesis −4 close parenthesis")[$8 − ( −4 )$] #solutionbox[ #figure(figph[This figure has a row of 12 light pink circles, representing positive counters. The first 8 are separated from the last 4. Below the last 4 is a row of 4 dark pink circles, representing negative counters. These four negative counters are circled.], alt: "This figure has a row of 12 light pink circles, representing positive counters. The first 8 are separated from the last 4. Below the last 4 is a row of 4 dark pink circles, representing negative counters. These four negative counters are circled.", caption: none) #linebreak() 12 ] #math.equation(block: true, alt: "7 minus open parenthesis −3 close parenthesis")[$7 − ( −3 )$] #strong[Simplify Expressions with Integers] In the following exercises, simplify each expression. + ⓐ #math.equation(block: false, alt: "15 minus 6")[$#h(0.2em) 15 − 6 #h(1em)$] + ⓑ #math.equation(block: false, alt: "15 plus open parenthesis −6 close parenthesis")[$#h(0.2em) 15 + ( −6 )$] #solutionbox[ + ⓐ 9 + ⓑ 9 ] + ⓐ #math.equation(block: false, alt: "12 minus 9")[$#h(0.2em) 12 − 9 #h(1em)$] + ⓑ #math.equation(block: false, alt: "12 plus open parenthesis −9 close parenthesis")[$#h(0.2em) 12 + ( −9 )$] + ⓐ #math.equation(block: false, alt: "44 minus 28")[$#h(0.2em) 44 − 28 #h(1em)$] + ⓑ #math.equation(block: false, alt: "44 plus open parenthesis −28 close parenthesis")[$#h(0.2em) 44 + ( −28 )$] #solutionbox[ + ⓐ 16 + ⓑ 16 ] + ⓐ #math.equation(block: false, alt: "35 minus 16")[$#h(0.2em) 35 − 16 #h(1em)$] + ⓑ #math.equation(block: false, alt: "35 plus open parenthesis −16 close parenthesis")[$35 + ( −16 )$] + ⓐ #math.equation(block: false, alt: "8 minus open parenthesis −9 close parenthesis")[$#h(0.2em) 8 − ( −9 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "8 plus 9")[$8 + 9$] #solutionbox[ + ⓐ 17 + ⓑ 17 ] + ⓐ #math.equation(block: false, alt: "4 minus open parenthesis −4 close parenthesis")[$#h(0.2em) 4 − ( −4 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "4 plus 4")[$#h(0.2em) 4 + 4$] + ⓐ #math.equation(block: false, alt: "27 minus open parenthesis −18 close parenthesis")[$#h(0.2em) 27 − ( −18 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "27 plus 18")[$#h(0.2em) 27 + 18$] #solutionbox[ + ⓐ 45 + ⓑ 45 ] + ⓐ #math.equation(block: false, alt: "46 minus open parenthesis −37 close parenthesis")[$#h(0.2em) 46 − ( −37 ) #h(1em)$] + ⓑ #math.equation(block: false, alt: "46 plus 37")[$#h(0.2em) 46 + 37$] In the following exercises, simplify each expression. #math.equation(block: true, alt: "15 minus open parenthesis −12 close parenthesis")[$15 − ( −12 )$] #solutionbox[ 27 ] #math.equation(block: true, alt: "14 minus open parenthesis −11 close parenthesis")[$14 − ( −11 )$] #math.equation(block: true, alt: "10 minus open parenthesis −19 close parenthesis")[$10 − ( −19 )$] #solutionbox[ 29 ] #math.equation(block: true, alt: "11 minus open parenthesis −18 close parenthesis")[$11 − ( −18 )$] #math.equation(block: true, alt: "48 minus 87")[$48 − 87$] #solutionbox[ −39 ] #math.equation(block: true, alt: "45 minus 69")[$45 − 69$] #math.equation(block: true, alt: "31 minus 79")[$31 − 79$] #solutionbox[ −48 ] #math.equation(block: true, alt: "39 minus 81")[$39 − 81$] #math.equation(block: true, alt: "−31 minus 11")[$−31 − 11$] #solutionbox[ −42 ] #math.equation(block: true, alt: "−32 minus 18")[$−32 − 18$] #math.equation(block: true, alt: "−17 minus 42")[$−17 − 42$] #solutionbox[ −59 ] #math.equation(block: true, alt: "−19 minus 46")[$−19 − 46$] #math.equation(block: true, alt: "−103 minus open parenthesis −52 close parenthesis")[$−103 − ( −52 )$] #solutionbox[ −51 ] #math.equation(block: true, alt: "−105 minus open parenthesis −68 close parenthesis")[$−105 − ( −68 )$] #math.equation(block: true, alt: "−45 minus open parenthesis −54 close parenthesis")[$−45 − ( −54 )$] #solutionbox[ 9 ] #math.equation(block: true, alt: "−58 minus open parenthesis −67 close parenthesis")[$−58 − ( −67 )$] #math.equation(block: true, alt: "8 minus 3 minus 7")[$8 − 3 − 7$] #solutionbox[ −2 ] #math.equation(block: true, alt: "9 minus 6 minus 5")[$9 − 6 − 5$] #math.equation(block: true, alt: "−5 minus 4 plus 7")[$−5 − 4 + 7$] #solutionbox[ −2 ] #math.equation(block: true, alt: "−3 minus 8 plus 4")[$−3 − 8 + 4$] #math.equation(block: true, alt: "−14 minus open parenthesis −27 close parenthesis plus 9")[$−14 − ( −27 ) + 9$] #solutionbox[ 22 ] #math.equation(block: true, alt: "−15 minus open parenthesis −28 close parenthesis plus 5")[$−15 − ( −28 ) + 5$] #math.equation(block: true, alt: "71 plus open parenthesis −10 close parenthesis minus 8")[$71 + ( −10 ) − 8$] #solutionbox[ 53 ] #math.equation(block: true, alt: "64 plus open parenthesis −17 close parenthesis minus 9")[$64 + ( −17 ) − 9$] #math.equation(block: true, alt: "−16 minus open parenthesis −4 plus 1 close parenthesis minus 7")[$−16 − ( −4 + 1 ) − 7$] #solutionbox[ −20 ] #math.equation(block: true, alt: "−15 minus open parenthesis −6 plus 4 close parenthesis minus 3")[$−15 − ( −6 + 4 ) − 3$] #math.equation(block: true, alt: "open parenthesis 2 minus 7 close parenthesis minus open parenthesis 3 minus 8 close parenthesis")[$( 2 − 7 ) − ( 3 − 8 )$] #solutionbox[ 0 ] #math.equation(block: true, alt: "open parenthesis 1 minus 8 close parenthesis minus open parenthesis 2 minus 9 close parenthesis")[$( 1 − 8 ) − ( 2 − 9 )$] #math.equation(block: true, alt: "minus open parenthesis 6 minus 8 close parenthesis minus open parenthesis 2 minus 4 close parenthesis")[$− ( 6 − 8 ) − ( 2 − 4 )$] #solutionbox[ 4 ] #math.equation(block: true, alt: "minus open parenthesis 4 minus 5 close parenthesis minus open parenthesis 7 minus 8 close parenthesis")[$− ( 4 − 5 ) − ( 7 − 8 )$] #math.equation(block: true, alt: "25 minus open bracket 10 minus open parenthesis 3 minus 12 close parenthesis close bracket")[$25 − [ 10 − ( 3 − 12 ) ]$] #solutionbox[ 6 ] #math.equation(block: true, alt: "32 minus open bracket 5 minus open parenthesis 15 minus 20 close parenthesis close bracket")[$32 − [ 5 − ( 15 − 20 ) ]$] #math.equation(block: true, alt: "6 times 3 minus 4 times 3 minus 7 times 2")[$6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2$] #solutionbox[ –8 ] #math.equation(block: true, alt: "5 times 7 minus 8 times 2 minus 4 times 9")[$5 ⋅ 7 − 8 ⋅ 2 − 4 ⋅ 9$] #math.equation(block: true, alt: "5 squared minus 6 squared")[$5^(2) − 6^(2)$] #solutionbox[ −11 ] #math.equation(block: true, alt: "6 squared minus 7 squared")[$6^(2) − 7^(2)$] #strong[Evaluate Variable Expressions with Integers] In the following exercises, evaluate each expression for the given values. #math.equation(block: true, alt: "x minus 6 when")[$x − 6 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3 #h(1em)$] + ⓑ #math.equation(block: false, alt: "x equals −3")[$#h(0.2em) x = −3$] #solutionbox[ + ⓐ −3 + ⓑ −9 ] #math.equation(block: true, alt: "x minus 4 when")[$x − 4 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "x equals 5")[$#h(0.2em) x = 5 #h(1em)$] + ⓑ #math.equation(block: false, alt: "x equals −5")[$#h(0.2em) x = −5$] #math.equation(block: true, alt: "5 minus y when")[$5 − y #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "y equals 2")[$#h(0.2em) y = 2 #h(1em)$] + ⓑ #math.equation(block: false, alt: "y equals −2")[$#h(0.2em) y = −2$] #solutionbox[ + ⓐ 3 + ⓑ 7 ] #math.equation(block: true, alt: "8 minus y when")[$8 − y #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "y equals 3")[$#h(0.2em) y = 3 #h(1em)$] + ⓑ #math.equation(block: false, alt: "y equals −3")[$#h(0.2em) y = −3$] #math.equation(block: true, alt: "4 x squared minus 15 x plus 1 when x equals 3")[$4 x^(2) − 15 x + 1 #h(0.2em) "when" #h(0.2em) x = 3$] #solutionbox[ −8 ] #math.equation(block: true, alt: "5 x squared minus 14 x plus 7 when x equals 2")[$5 x^(2) − 14 x + 7 #h(0.2em) "when" #h(0.2em) x = 2$] #math.equation(block: true, alt: "−12 minus 5 x squared when x equals 6")[$−12 − 5 x^(2) #h(0.2em) "when" #h(0.2em) x = 6$] #solutionbox[ −192 ] #math.equation(block: true, alt: "−19 minus 4 x squared when x equals 5")[$−19 − 4 x^(2) #h(0.2em) "when" #h(0.2em) x = 5$] #strong[Translate Word 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: "3")[$3$] and #math.equation(block: false, alt: "−10")[$−10$] + ⓑ Subtract #math.equation(block: false, alt: "−20")[$−20$] from #math.equation(block: false, alt: "45")[$45$] #solutionbox[ + ⓐ 3 − (−10) = 13 + ⓑ 45 − (−20) = 65 ] + ⓐ The difference of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "−12")[$−12$] + ⓑ Subtract #math.equation(block: false, alt: "−13")[$−13$] from #math.equation(block: false, alt: "50")[$50$] + ⓐ The difference of #math.equation(block: false, alt: "−6")[$−6$] and #math.equation(block: false, alt: "9")[$9$] + ⓑ Subtract #math.equation(block: false, alt: "−12")[$−12$] from #math.equation(block: false, alt: "−16")[$−16$] #solutionbox[ + ⓐ −6 − 9 = −15 + ⓑ −16 − (−12) = −4 ] + ⓐ The difference of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "9")[$9$] + ⓑ Subtract #math.equation(block: false, alt: "minus 15")[$− 15$] from #math.equation(block: false, alt: "minus 19")[$− 19$] + ⓐ #math.equation(block: false, alt: "8")[$#h(0.2em) 8$] less than #math.equation(block: false, alt: "−17")[$−17$] + ⓑ #math.equation(block: false, alt: "minus 24")[$#h(0.2em) − 24$] minus #math.equation(block: false, alt: "37")[$37$] #solutionbox[ + ⓐ −17 − 8 = −25 + ⓑ −24 − 37 = −61 ] + ⓐ #math.equation(block: false, alt: "5")[$#h(0.2em) 5$] less than #math.equation(block: false, alt: "−14")[$−14$] + ⓑ #math.equation(block: false, alt: "−13")[$#h(0.2em) −13$] minus #math.equation(block: false, alt: "42")[$42$] + ⓐ #math.equation(block: false, alt: "21")[$#h(0.2em) 21$] less than#math.equation(block: false, alt: "6")[$#h(0.2em) 6$] + ⓑ #math.equation(block: false, alt: "31")[$#h(0.2em) 31$] subtracted from #math.equation(block: false, alt: "−19")[$−19$] #solutionbox[ + ⓐ 6 − 21 = −15 + ⓑ −19 − 31 = −50 ] + ⓐ #math.equation(block: false, alt: "34")[$#h(0.2em) 34$] less than#math.equation(block: false, alt: "7")[$#h(0.2em) 7$] + ⓑ #math.equation(block: false, alt: "29")[$#h(0.2em) 29$] subtracted from #math.equation(block: false, alt: "−50")[$−50$] #strong[Subtract Integers in Applications] In the following exercises, solve the following applications. #strong[Temperature] One morning, the temperature in Urbana, Illinois, was #math.equation(block: false, alt: "28° Fahrenheit.")[$"28° Fahrenheit."$] By evening, the temperature had dropped #math.equation(block: false, alt: "38° Fahrenheit.")[$"38° Fahrenheit."$] What was the temperature that evening? #solutionbox[ −10° ] #strong[Temperature] On Thursday, the temperature in Spincich Lake, Michigan, was #math.equation(block: false, alt: "22° Fahrenheit.")[$"22° Fahrenheit."$] By Friday, the temperature had dropped #math.equation(block: false, alt: "35° Fahrenheit.")[$"35° Fahrenheit."$] What was the temperature on Friday? #strong[Temperature] On January 15, the high temperature in Anaheim, California, was #math.equation(block: false, alt: "84° Fahrenheit.")[$"84° Fahrenheit."$] That same day, the high temperature in Embarrass, Minnesota was #math.equation(block: false, alt: "−12° Fahrenheit.")[$"−12° Fahrenheit."$] What was the difference between the temperature in Anaheim and the temperature in Embarrass? #solutionbox[ 96° ] #strong[Temperature] On January 21, the high temperature in Palm Springs, California, was #math.equation(block: false, alt: "89°,")[$"89°,"$] and the high temperature in Whitefield, New Hampshire was #math.equation(block: false, alt: "−31° .")[$"−31°" .$] What was the difference between the temperature in Palm Springs and the temperature in Whitefield? #strong[Football] At the first down, the Warriors football team had the ball on their #math.equation(block: false, alt: "30-yard line.")[$"30-yard line."$] On the next three downs, they gained #math.equation(block: false, alt: "2 yards,")[$"2 yards,"$] lost #math.equation(block: false, alt: "7 yards,")[$"7 yards,"$] and lost #math.equation(block: false, alt: "4 yards.")[$"4 yards."$] What was the yard line at the end of the third down? #solutionbox[ 21-yard line ] #strong[Football] At the first down, the Barons football team had the ball on their #math.equation(block: false, alt: "20-yard line.")[$"20-yard line."$] On the next three downs, they lost #math.equation(block: false, alt: "8 yards,")[$"8 yards,"$] gained #math.equation(block: false, alt: "5 yards,")[$"5 yards,"$] and lost #math.equation(block: false, alt: "6 yards.")[$"6 yards."$] What was the yard line at the end of the third down? #strong[Checking Account] John has #math.equation(block: false, alt: "$148")[$"$148"$] in his checking account. He writes a check for #math.equation(block: false, alt: "$83.")[$"$83."$] What is the new balance in his checking account? #solutionbox[ \$65 ] #strong[Checking Account] Ellie has #math.equation(block: false, alt: "$426")[$"$426"$] in her checking account. She writes a check for #math.equation(block: false, alt: "$152.")[$"$152."$] What is the new balance in her checking account? #strong[Checking Account] Gina has #math.equation(block: false, alt: "$210")[$"$210"$] in her checking account. She writes a check for #math.equation(block: false, alt: "$250.")[$"$250."$] What is the new balance in her checking account? #solutionbox[ −\$40 ] #strong[Checking Account] Frank has #math.equation(block: false, alt: "$94")[$"$94"$] in his checking account. He writes a check for #math.equation(block: false, alt: "$110.")[$"$110."$] What is the new balance in his checking account? #strong[Checking Account] Bill has a balance of #math.equation(block: false, alt: "−$14")[$"−$14"$] in his checking account. He deposits #math.equation(block: false, alt: "$40")[$"$40"$] to the account. What is the new balance? #solutionbox[ \$26 ] #strong[Checking Account] Patty has a balance of #math.equation(block: false, alt: "−$23")[$"−$23"$] in her checking account. She deposits #math.equation(block: false, alt: "$80")[$"$80"$] to the account. What is the new balance? === Everyday Math #strong[Camping] Rene is on an Alpine hike. The temperature is#math.equation(block: false, alt: "minus 7 ° .")[$− "7" "°" .$] Rene’s sleeping bag is rated “comfortable to #math.equation(block: false, alt: "minus 20 °”.")[$− "20" "°”."$] How much can the temperature change before it is too cold for Rene’s sleeping bag? #solutionbox[ 13° ] #strong[Scuba Diving] Shelly’s scuba watch is guaranteed to be watertight to #math.equation(block: false, alt: "−100 feet .")[$−100 #h(0.2em) "feet" .$] She is diving at #math.equation(block: false, alt: "−45 feet")[$−45 #h(0.2em) "feet"$] on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed? === Writing Exercises Explain why the difference of #math.equation(block: false, alt: "9")[$9$] and #math.equation(block: false, alt: "−6")[$−6$] is #math.equation(block: false, alt: "15.")[$15 .$] #solutionbox[ Sample answer: On a number line, 9 is 15 units away from −6. ] Why is the result of subtracting #math.equation(block: false, alt: "3 minus open parenthesis −4 close parenthesis")[$3 − ( −4 )$] the same as the result of adding #math.equation(block: false, alt: "3 plus 4 ?")[$3 + 4 ?$] === 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 mathematics skills related to integers. The table lists five 'I can...' statements (e.g., model subtraction of integers) and three columns for assessment: Confidently, With some help, and No-I don't get it!], alt: "A self-assessment checklist for mathematics skills related to integers. The table lists five 'I can...' statements (e.g., model subtraction of integers) and three columns for assessment: Confidently, With some help, and No-I don't get it!", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?