#set document(title: "1.3 Add and 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")) == 1.3#h(0.6em)Add and Subtract Integers #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A more thorough introduction to the topics covered in this section can be found in the #emph[Prealgebra] chapter, #strong[Integers]. ] === Use Negatives and Opposites Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. #strong[Negative numbers] are numbers less than #math.equation(block: false, alt: "0 .")[$0 .$] The #strong[negative numbers] are to the left of zero on the number line. #figure(figph[A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.], alt: "A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.", caption: [The number line shows the location of positive and negative numbers.]) The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number. Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative. Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. #figure(figph[A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.], alt: "A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.", caption: [The numbers on a number line increase in value going from left to right and decrease in value going from right to left.]) Remember that we use the notation: #emph[a] \< #emph[b] (read “a is less than b”) when a is to the left of b on the number line. #emph[a] \> #emph[b] (read “#emph[a] is greater than #emph[b]”) when #emph[a] is to the right of #emph[b] on the number line. Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in are called the integers. The integers are the numbers #math.equation(block: false, alt: "… minus 3 , −2 , −1 , 0 , 1 , 2 , 3 …")[$"…" − 3 , −2 , −1 , 0 , 1 , 2 , 3 "…"$] #figure(figph[A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.], alt: "A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.", caption: [All the marked numbers are called #emph[integers].]) #examplebox("Example 1")[][ Order each of the following pairs of numbers, using \< or \>: ⓐ #math.equation(block: false, alt: "14 ___ 6")[$14 \_\_\_ 6$] ⓑ #math.equation(block: false, alt: "−1 ___ 9")[$−1 \_\_\_ 9$] ⓒ #math.equation(block: false, alt: "−1 ___ −4")[$−1 \_\_\_ −4$] ⓓ #math.equation(block: false, alt: "2 ___ −20 .")[$2 \_\_\_ −20 .$] #solutionbox[ It may be helpful to refer to the number line shown. #linebreak() #figure(figph[A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.], alt: "A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.", caption: none) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() 14 is to the right of 6 on the number line.], [#math.equation(block: false, alt: "14 ___ 6; 14 greater than 6")[$14 \_\_\_ 6 \ 14 > 6$]]), [ⓑ #linebreak() −1 is to the left of 9 on the number line.], [#math.equation(block: false, alt: "−1 ___ 9; −1 less than 9")[$−1 \_\_\_ 9 \ −1 < 9$]], [ⓒ #linebreak() −1 is to the right of −4 on the number line.], [#math.equation(block: false, alt: "−1 ___ −4; −1 greater than −4")[$−1 \_\_\_ −4 \ −1 > −4$]], [ⓓ #linebreak() 2 is to the right of −20 on the number line.], [#math.equation(block: false, alt: "2 ___ −20; 2 greater than −20")[$2 \_\_\_ −20 \ 2 > −20$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than")[$<$] or #math.equation(block: false, alt: "greater than :")[$> ":"$] ⓐ #math.equation(block: false, alt: "15 ___ 7")[$15 \_\_\_ 7$] ⓑ #math.equation(block: false, alt: "−2 ___ 5")[$−2 \_\_\_ 5$] ⓒ #math.equation(block: false, alt: "−3 ___ −7")[$−3 \_\_\_ −7$] #linebreak() ⓓ #math.equation(block: false, alt: "5 ___ −17 .")[$5 \_\_\_ −17 .$] #solutionbox[ ⓐ \> ⓑ \< ⓒ \> ⓓ \> ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Order each of the following pairs of numbers, using #math.equation(block: false, alt: "less than")[$<$] or #math.equation(block: false, alt: "greater than :")[$> ":"$] ⓐ #math.equation(block: false, alt: "8 ___ 13")[$8 \_\_\_ 13$] ⓑ #math.equation(block: false, alt: "3 ___ −4")[$3 \_\_\_ −4$] ⓒ #math.equation(block: false, alt: "−5 ___ −2")[$−5 \_\_\_ −2$] #linebreak() ⓓ #math.equation(block: false, alt: "9 ___ −21 .")[$9 \_\_\_ −21 .$] #solutionbox[ ⓐ \< ⓑ \> ⓒ \< ⓓ \> ] ] You may have noticed that, on the #strong[number line], the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and #math.equation(block: false, alt: "−2")[$−2$] are the same distance from zero, they are called #strong[opposite]#strong[s]. The opposite of 2 is #math.equation(block: false, alt: "−2 ,")[$−2 ,$] and the opposite of #math.equation(block: false, alt: "−2")[$−2$] is 2. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Opposite] The #strong[opposite] of a number is the number that is the same distance from zero on the number line but on the opposite side of zero. ] illustrates the definition. #figure(figph[A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.], alt: "A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.", caption: [The opposite of 3 is #math.equation(block: false, alt: "−3 .")[$−3 .$]]) Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways. #math.equation(block: true, alt: "10 minus 4, Between two numbers, it indicates the operation of subtraction .; We read 10 minus 4 as “ 10 minus 4. ”; −8, In front of a number, it indicates a negative number.; We read −8 as “negative eight.”; − x, In front of a variable, it indicates the opposite . We read − x as “the opposite of x . ”; − open parenthesis −2 close parenthesis, Here there are two “ minus ” signs. The one in the parentheses tells us the number is; negative 2 . The one outside the parentheses tells us to take the opposite of −2 .; We read − open parenthesis −2 close parenthesis as “the opposite of negative two.”")[$10 − 4 & & & "Between two numbers, it indicates the operation of" #h(0.2em) "subtraction" . \ & & & "We read" #h(0.2em) 10 − 4 #h(0.2em) "as" #h(0.2em) "“" 10 #h(0.2em) "minus" #h(0.2em) 4. "”" \ −8 & & & "In front of a number, it indicates a" #h(0.2em) "negative" #h(0.2em) "number." \ & & & "We read" #h(0.2em) −8 #h(0.2em) "as “negative eight.”" \ "−" x & & & "In front of a variable, it indicates the" #h(0.2em) "opposite" . #h(0.2em) "We read" #h(0.2em) "−" x #h(0.2em) "as “the opposite of" #h(0.2em) x . "”" \ "−" ( −2 ) & & & "Here there are two" #h(0.2em) "“" − "”" #h(0.2em) "signs. The one in the parentheses tells us the number is" \ & & & "negative" #h(0.2em) 2 . #h(0.2em) "The one outside the parentheses tells us to take the" #h(0.2em) "opposite" #h(0.2em) "of" #h(0.2em) −2 . \ & & & "We read" #h(0.2em) "−" ( −2 ) #h(0.2em) "as “the opposite of negative two.”"$]#figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "10 minus 4")[$10 − 4$]], [Between two numbers, it indicates the operation of #emph[subtraction]. #linebreak() We read #math.equation(block: false, alt: "10 minus 4")[$10 − 4$] as "10 minus 4."]), [#math.equation(block: false, alt: "−8")[$−8$]], [In front of a number, it indicates a #emph[negative] number. #linebreak() We read −8 as "negative eight."], [#math.equation(block: false, alt: "− x")[$"−" x$]], [In front of a variable, it indicates the #emph[opposite]. We read #math.equation(block: false, alt: "− x")[$"−" x$] as "the opposite of #math.equation(block: false, alt: "x")[$x$]."], [#math.equation(block: false, alt: "− open parenthesis −2 close parenthesis")[$"−" ( −2 )$]], [Here there are two "−" signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the #emph[opposite] of −2. #linebreak() We read #math.equation(block: false, alt: "− open parenthesis −2 close parenthesis")[$"−" ( −2 )$] as "the opposite of negative two."], )) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Opposite Notation] #math.equation(block: false, alt: "− a")[$"−" a$] means the opposite of the number #emph[a]. The notation #math.equation(block: false, alt: "− a")[$"−" a$] is read as “the opposite of #emph[a].” ] #examplebox("Example 2")[][ Find: ⓐ the opposite of 7 ⓑ the opposite of #math.equation(block: false, alt: "−10")[$−10$] ⓒ #math.equation(block: false, alt: "− open parenthesis −6 close parenthesis .")[$"−" ( −6 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ −7 is the same distance from 0 as 7, but on the opposite side of 0.], [#figure(figph[A number line illustrates the distance from 0. Points at -7 and 7 are marked. Brackets above show that the distance from -7 to 0 is 7 units, and the distance from 0 to 7 is 7 units.], alt: "A number line illustrates the distance from 0. Points at -7 and 7 are marked. Brackets above show that the distance from -7 to 0 is 7 units, and the distance from 0 to 7 is 7 units.", caption: none) #linebreak() The opposite of 7 is −7.]), [ⓑ 10 is the same distance from 0 as −10, but on the opposite side of 0.], [#figure(figph[A number line shows points at -10, 0, and 10. Arcs indicate a distance of 10 units from -10 to 0, and another 10 units from 0 to 10.], alt: "A number line shows points at -10, 0, and 10. Arcs indicate a distance of 10 units from -10 to 0, and another 10 units from 0 to 10.", caption: none) #linebreak() The opposite of −10 is 10.], [ⓒ −(−6) (the opposite of –6).], [#figure(figph[A number line illustrating absolute value, showing points at -6, 0, and 6. The distance from 0 to -6 is 6, and the distance from 0 to 6 is also 6.], alt: "A number line illustrating absolute value, showing points at -6, 0, and 6. The distance from 0 to -6 is 6, and the distance from 0 to 6 is also 6.", caption: none) #linebreak() The opposite of −(−6) is 6.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find: ⓐ the opposite of 4 ⓑ the opposite of #math.equation(block: false, alt: "−3")[$−3$] ⓒ #math.equation(block: false, alt: "− open parenthesis −1 close parenthesis .")[$"−" ( −1 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ 3 ⓒ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find: ⓐ the opposite of 8 ⓑ the opposite of #math.equation(block: false, alt: "−5")[$−5$] ⓒ #math.equation(block: false, alt: "− open parenthesis −5 close parenthesis .")[$"−" ( −5 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−8")[$−8$] ⓑ 5 ⓒ 5 ] ] Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the #strong[integers]. The integers are the numbers #math.equation(block: false, alt: "… minus 3 , −2 , −1 , 0 , 1 , 2 , 3 …")[$"…" − 3 , −2 , −1 , 0 , 1 , 2 , 3 "…"$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Integers] The whole numbers and their opposites are called the #strong[integers]. The integers are the numbers #math.equation(block: true, alt: "… minus 3 , −2 , −1 , 0 , 1 , 2 , 3 …")[$"…" − 3 , −2 , −1 , 0 , 1 , 2 , 3 "…"$] ] When evaluating the opposite of a #strong[variable], we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether #math.equation(block: false, alt: "− x")[$"−" x$] is positive or negative. We can see this in . #examplebox("Example 3")[][ Evaluate ⓐ #math.equation(block: false, alt: "− x ,")[$"−" x ,$] when #math.equation(block: false, alt: "x equals 8")[$x = 8$] ⓑ #math.equation(block: false, alt: "− x ,")[$"−" x ,$] when #math.equation(block: false, alt: "x equals −8 .")[$x = −8 .$] #solutionbox[ + ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#figure(figph[The image states that to evaluate an expression when x = 8 means to substitute the number 8 in place of the variable x.], alt: "The image states that to evaluate an expression when x = 8 means to substitute the number 8 in place of the variable x.", caption: none)], []), [], [−#emph[x]], [#figure(figph[The text on a white background states, 'Substitute 8 for x.'], alt: "The text on a white background states, 'Substitute 8 for x.'", caption: none)], [#figure(figph[A mathematical expression showing a negative sign followed by the number 8 enclosed in parentheses, appearing as -(8) in red against a white background.], alt: "A mathematical expression showing a negative sign followed by the number 8 enclosed in parentheses, appearing as -(8) in red against a white background.", caption: none)], [Write the opposite of 8.], [#figure(figph[The image displays the number -8, rendered in a digital, sans-serif font. The negative sign is a horizontal line preceding the numeral '8', which is depicted with two closed loops stacked vertically.], alt: "The image displays the number -8, rendered in a digital, sans-serif font. The negative sign is a horizontal line preceding the numeral '8', which is depicted with two closed loops stacked vertically.", caption: none)], )) #linebreak() #linebreak() + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#figure(figph[The image shows the mathematical instruction: 'To evaluate when x = -8 means to substitute -8 for -x.' The number -8 is highlighted in a reddish color.], alt: "The image shows the mathematical instruction: 'To evaluate when x = -8 means to substitute -8 for -x.' The number -8 is highlighted in a reddish color.", caption: none)], []), [], [−#emph[x]], [#figure(figph[The image shows a mathematical instruction: 'Substitute -8 for x.' The text is gray, with '-8' highlighted in red.], alt: "The image shows a mathematical instruction: 'Substitute -8 for x.' The text is gray, with '-8' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression shows the negation of negative eight, written as -(-8), with the '8' and minus sign inside the parentheses highlighted in a reddish hue against a white background.], alt: "A mathematical expression shows the negation of negative eight, written as -(-8), with the '8' and minus sign inside the parentheses highlighted in a reddish hue against a white background.", caption: none)], [Write the opposite of −8.], [8], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "− n ,")[$"−" n ,$] when ⓐ #math.equation(block: false, alt: "n equals 4")[$n = 4$] ⓑ #math.equation(block: false, alt: "n equals −4 .")[$n = −4 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ 4 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "− m ,")[$"−" m ,$] when ⓐ #math.equation(block: false, alt: "m equals 11")[$m = 11$] ⓑ #math.equation(block: false, alt: "m equals −11 .")[$m = −11 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−11")[$−11$] ⓑ 11 ] ] === Simplify: Expressions with Absolute Value We saw that numbers such as #math.equation(block: false, alt: "2 and −2")[$2 #h(0.2em) "and" #h(0.2em) −2$] are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the #strong[absolute value] of that number. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Absolute Value] The #strong[absolute value] of a number is its distance from 0 on the number line. The absolute value of a number #emph[n] is written as #math.equation(block: false, alt: "| n | .")[$| n | .$] ] For example, - #math.equation(block: false, alt: "−5 is 5")[$−5 #h(0.2em) "is" #h(0.2em) 5$] units away from #math.equation(block: false, alt: "0 ,")[$0 ,$] so #math.equation(block: false, alt: "| −5 | equals 5 .")[$| −5 | = 5 .$] - #math.equation(block: false, alt: "5 is 5")[$#h(0.65em) 5 #h(0.2em) "is" #h(0.2em) 5$] units away from #math.equation(block: false, alt: "0 ,")[$0 ,$] so #math.equation(block: false, alt: "| 5 | equals 5 .")[$| 5 | = 5 .$] illustrates this idea. #figure(figph[A number line is shown ranging from negative 5 to 5. A bracket labeled “5 units” lies above the points negative 5 to 0. An arrow labeled “negative 5 is 5 units from 0, so absolute value of negative 5 equals 5.” is written above the labeled bracket. A bracket labeled “5 units” lies above the points “0” to “5”. An arrow labeled “5 is 5 units from 0, so absolute value of 5 equals 5.” and is written above the labeled bracket.], alt: "A number line is shown ranging from negative 5 to 5. A bracket labeled “5 units” lies above the points negative 5 to 0. An arrow labeled “negative 5 is 5 units from 0, so absolute value of negative 5 equals 5.” is written above the labeled bracket. A bracket labeled “5 units” lies above the points “0” to “5”. An arrow labeled “5 is 5 units from 0, so absolute value of 5 equals 5.” and is written above the labeled bracket.", caption: [The integers #math.equation(block: false, alt: "5 and are 5")[$5 #h(0.2em) "and are" #h(0.2em) 5$] units away from #math.equation(block: false, alt: "0 .")[$0 .$]]) The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from #math.equation(block: false, alt: "0 to 0")[$0 #h(0.2em) "to" #h(0.2em) 0$] on the number line is zero units. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Property of Absolute Value] #math.equation(block: false, alt: "| n | greater than or equal to 0")[$| n | ≥ 0$] for all numbers Absolute values are always greater than or equal to zero! ] Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero. #examplebox("Example 4")[][ Simplify: ⓐ #math.equation(block: false, alt: "| 3 |")[$| 3 |$] ⓑ #math.equation(block: false, alt: "| −44 |")[$| −44 |$] ⓒ #math.equation(block: false, alt: "| 0 |")[$| 0 |$]. #solutionbox[ The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative. ⓐ #math.equation(block: false, alt: "| 3 |")[$| 3 |$] #linebreak() #math.equation(block: false, alt: "3")[$#h(1.2em) 3$] #linebreak() #linebreak() ⓑ #math.equation(block: false, alt: "| −44 |")[$| −44 |$] #linebreak() #math.equation(block: false, alt: "44")[$#h(1.5em) 44$] #linebreak() #linebreak() ⓒ #math.equation(block: false, alt: "| 0 |")[$| 0 |$] #linebreak() #math.equation(block: false, alt: "0")[$#h(1.2em) 0$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "| 4 |")[$| 4 |$] ⓑ #math.equation(block: false, alt: "| −28 |")[$| −28 |$] ⓒ #math.equation(block: false, alt: "| 0 | .")[$| 0 | .$] #solutionbox[ ⓐ 4 ⓑ 28 ⓒ 0 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "| −13 |")[$| −13 |$] ⓑ #math.equation(block: false, alt: "| 47 |")[$| 47 |$] ⓒ #math.equation(block: false, alt: "| 0 | .")[$| 0 | .$] #solutionbox[ ⓐ 13 ⓑ 47 ⓒ 0 ] ] In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers! #examplebox("Example 5")[][ Fill in #math.equation(block: false, alt: "less than , greater than , or equals")[$< , > , #h(0.2em) "or" #h(0.2em) =$] for each of the following pairs of numbers: ⓐ #math.equation(block: false, alt: "| −5 | ___ minus | −5 |")[$| −5 | \_\_\_ − | −5 |$] ⓑ #math.equation(block: false, alt: "8 ___ minus | −8 |")[$8 \_\_\_ − | −8 |$] ⓒ #math.equation(block: false, alt: "−9 ___ minus | −9 |")[$−9 \_\_\_ − | −9 |$] ⓓ #math.equation(block: false, alt: "− open parenthesis −16 close parenthesis ___ minus | −16 |")[$"−" ( −16 ) \_\_\_ − | −16 |$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() Simplify. #linebreak() Order. #linebreak()], [#math.equation(block: false, alt: "| −5 |, ___, minus | −5 |; 5, ___, minus 5; 5, greater than, −5; | −5 |, greater than, minus | −5 |")[$| −5 | & \_\_\_ & − | −5 | \ 5 & \_\_\_ & − 5 \ 5 & > & −5 \ | −5 | & > & − | −5 |$]]), [ⓑ #linebreak() Simplify. #linebreak() Order. #linebreak()], [#math.equation(block: false, alt: "8, ___, minus | −8 |; 8, ___, minus 8; 8, greater than, −8; 8, greater than, minus | −8 |")[$8 & \_\_\_ & − | −8 | \ 8 & \_\_\_ & − 8 \ 8 & > & −8 \ 8 & > & − | −8 |$]], [ⓒ #linebreak() Simplify. #linebreak() Order. #linebreak()], [#math.equation(block: false, alt: "−9, ___, minus | −9 |; −9, ___, minus 9; −9, equals, −9; −9, equals, minus | −9 |")[$−9 & \_\_\_ & − | −9 | \ −9 & \_\_\_ & − 9 \ −9 & = & −9 \ −9 & = & − | −9 |$]], [ⓓ #linebreak() Simplify. #linebreak() Order. #linebreak()], [#math.equation(block: false, alt: "− open parenthesis −16 close parenthesis, ___, minus | −16 |; 16, ___, minus 16; 16, greater than, −16; − open parenthesis −16 close parenthesis, greater than, minus | −16 |")[$"−" ( −16 ) & \_\_\_ & − | −16 | \ 16 & \_\_\_ & − 16 \ 16 & > & −16 \ "−" ( −16 ) & > & − | −16 |$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Fill in \<, \>, or #math.equation(block: false, alt: "equals")[$=$] for each of the following pairs of numbers: ⓐ #math.equation(block: false, alt: "| −9 | ___ minus | −9 |")[$| −9 | \_\_\_ − | −9 |$] ⓑ #math.equation(block: false, alt: "2 ___ minus | −2 |")[$2 \_\_\_ − | −2 |$] ⓒ #math.equation(block: false, alt: "−8 ___ | −8 |")[$−8 \_\_\_ | −8 |$] #linebreak() ⓓ #math.equation(block: false, alt: "− open parenthesis −9 close parenthesis ___ minus | −9 | .")[$"−" ( −9 ) \_\_\_ − | −9 | .$] #solutionbox[ ⓐ \> ⓑ \> ⓒ \< ⓓ \> ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Fill in \<, \>, or #math.equation(block: false, alt: "equals")[$=$] for each of the following pairs of numbers: ⓐ #math.equation(block: false, alt: "7 ___ minus | −7 |")[$7 \_\_\_ − | −7 |$] ⓑ #math.equation(block: false, alt: "− open parenthesis −10 close parenthesis ___ minus | −10 |")[$"−" ( −10 ) \_\_\_ − | −10 |$] #linebreak() ⓒ #math.equation(block: false, alt: "| −4 | ___ minus | −4 |")[$| −4 | \_\_\_ − | −4 |$] ⓓ #math.equation(block: false, alt: "−1 ___ | −1 | .")[$−1 \_\_\_ | −1 | .$] #solutionbox[ ⓐ \> ⓑ \> ⓒ \> ⓓ \< ] ] We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the #strong[absolute value] of the resulting number. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Grouping Symbols] #math.equation(block: true, alt: "Parentheses, open parenthesis close parenthesis, Braces, { }; Brackets, [ ], Absolute value, | |")[$"Parentheses" & & & & & & "Braces" & \{ \} \ "Brackets" & [ ] & & & & & "Absolute value" & #h(0.15em) | #h(0.2em) |$] ] In the next example, we simplify the expressions inside absolute value bars first, just as we do with parentheses. #examplebox("Example 6")[][ Simplify: #math.equation(block: false, alt: "24 minus | 19 minus 3 open parenthesis 6 minus 2 close parenthesis | .")[$24 − | 19 − 3 ( 6 − 2 ) | .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "24 minus | 19 minus 3 open parenthesis 6 minus 2 close parenthesis |")[$24 − | 19 − 3 ( 6 − 2 ) |$]]), [Work inside parentheses first: subtract 2 from 6.], [#math.equation(block: false, alt: "24 minus | 19 minus 3 open parenthesis 4 close parenthesis |")[$24 − | 19 − 3 ( 4 ) |$]], [Multiply 3(4).], [#math.equation(block: false, alt: "24 minus | 19 minus 12 |")[$24 − | 19 − 12 |$]], [Subtract inside the absolute value bars.], [#math.equation(block: false, alt: "24 minus | 7 |")[$24 − | 7 |$]], [Take the absolute value.], [#math.equation(block: false, alt: "24 minus 7")[$24 − 7$]], [Subtract.], [#math.equation(block: false, alt: "17")[$17$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "19 minus | 11 minus 4 open parenthesis 3 minus 1 close parenthesis | .")[$19 − | 11 − 4 ( 3 − 1 ) | .$] #solutionbox[ 16 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "9 minus | 8 minus 4 open parenthesis 7 minus 5 close parenthesis | .")[$9 − | 8 − 4 ( 7 − 5 ) | .$] #solutionbox[ 9 ] ] #examplebox("Example 7")[][ Evaluate: ⓐ #math.equation(block: false, alt: "| x | when x equals −35")[$| x | #h(0.2em) "when" #h(0.2em) x = −35$] ⓑ #math.equation(block: false, alt: "| − y | when y equals −20")[$| "−" y | #h(0.2em) "when" #h(0.2em) y = −20$] ⓒ #math.equation(block: false, alt: "− | u | when u equals 12")[$"−" | u | #h(0.2em) "when" #h(0.2em) u = 12$] ⓓ #math.equation(block: false, alt: "− | p | when p equals −14 .")[$"−" | p | #h(0.2em) "when" #h(0.2em) p = −14 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "| x | when x equals −35")[$| x | #h(0.2em) "when" #h(0.2em) x = −35$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "| x |")[$| x |$]]), [#figure(figph[The image shows the text 'Substitute -35 for x.' The word 'Substitute' is in black, and the number '-35' is in red, while 'for x.' is in black again. This text represents a mathematical instruction to replace the variable x with the value -35.], alt: "The image shows the text 'Substitute -35 for x.' The word 'Substitute' is in black, and the number '-35' is in red, while 'for x.' is in black again. This text represents a mathematical instruction to replace the variable x with the value -35.", caption: none)], [#figure(figph[The image displays the mathematical notation for the absolute value of negative thirty-five: | -35 |.], alt: "The image displays the mathematical notation for the absolute value of negative thirty-five: | -35 |.", caption: none)], [Take the absolute value.], [35], )) #linebreak() ⓑ #math.equation(block: false, alt: "| − y | when y equals −20")[$| "−" y | #h(0.2em) "when" #h(0.2em) y = −20$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "| minus y |")[$| − y |$]]), [#figure(figph[Substitute -20 for y.], alt: "Substitute -20 for y.", caption: none)], [#figure(figph[The absolute value of negative twenty, prefixed by a negative sign.], alt: "The absolute value of negative twenty, prefixed by a negative sign.", caption: none)], [Simplify.], [#math.equation(block: false, alt: "| 20 |")[$| 20 |$]], [Take the absolute value.], [20], )) #linebreak() ⓒ #math.equation(block: false, alt: "− | u | when u equals 12")[$"−" | u | #h(0.2em) "when" #h(0.2em) u = 12$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "minus | u |")[$− | u |$]]), [#figure(figph[The image displays the instruction: 'Substitute 12 for u.' in black text on a white background, with the number 12 highlighted in red.], alt: "The image displays the instruction: 'Substitute 12 for u.' in black text on a white background, with the number 12 highlighted in red.", caption: none)], [#figure(figph[A mathematical expression showing the negative of the absolute value of 12, represented as -| |12| |, which simplifies to -12.], alt: "A mathematical expression showing the negative of the absolute value of 12, represented as -| |12| |, which simplifies to -12.", caption: none)], [Take the absolute value.], [#math.equation(block: false, alt: "minus 12")[$− 12$]], )) #linebreak() ⓓ #math.equation(block: false, alt: "− | p | when p equals −14")[$"−" | p | #h(0.2em) "when" #h(0.2em) p = −14$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "minus | p |")[$− | p |$]]), [#figure(figph[Substitute -14 for p.], alt: "Substitute -14 for p.", caption: none)], [#figure(figph[A mathematical expression showing the absolute value of negative fourteen, written as |-14|, against a white background.], alt: "A mathematical expression showing the absolute value of negative fourteen, written as |-14|, against a white background.", caption: none)], [Take the absolute value.], [#math.equation(block: false, alt: "minus 14")[$− 14$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: ⓐ #math.equation(block: false, alt: "| x | when x equals −17")[$| x | #h(0.2em) "when" #h(0.2em) x = −17$] ⓑ #math.equation(block: false, alt: "| − y | when y equals −39")[$| "−" y | #h(0.2em) "when" #h(0.2em) y = −39$] ⓒ #math.equation(block: false, alt: "− | m | when m equals 22")[$"−" | m | #h(0.2em) "when" #h(0.2em) m = 22$] ⓓ #math.equation(block: false, alt: "− | p | when p equals −11 .")[$"−" | p | #h(0.2em) "when" #h(0.2em) p = −11 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "17")[$17$] ⓑ #math.equation(block: false, alt: "39")[$39$] ⓒ #math.equation(block: false, alt: "−22")[$−22$] ⓓ #math.equation(block: false, alt: "−11")[$−11$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: ⓐ #math.equation(block: false, alt: "| y | when y equals −23")[$| y | #h(0.2em) "when" #h(0.2em) y = −23$] ⓑ #math.equation(block: false, alt: "| − y | when y equals −21")[$| "−" y | #h(0.2em) "when" #h(0.2em) y = −21$] ⓒ #math.equation(block: false, alt: "− | n | when n equals 37")[$"−" | n | #h(0.2em) "when" #h(0.2em) n = 37$] ⓓ #math.equation(block: false, alt: "− | q | when q equals −49 .")[$"−" | q | #h(0.2em) "when" #h(0.2em) q = −49 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "23")[$23$] ⓑ #math.equation(block: false, alt: "21")[$21$] ⓒ #math.equation(block: false, alt: "−37")[$−37$] ⓓ #math.equation(block: false, alt: "−49")[$−49$] ] ] === Add Integers Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging. We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules. We let one color (blue) represent positive. The other color (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero. #figure(figph[In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.], alt: "In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.", caption: none) We will use the counters to show how to add the four addition facts using the numbers #math.equation(block: false, alt: "5 , −5")[$5 , −5$] and #math.equation(block: false, alt: "3 , −3 .")[$3 , −3 .$] #math.equation(block: true, alt: "5 plus 3, −5 plus open parenthesis −3 close parenthesis, −5 plus 3, 5 plus open parenthesis −3 close parenthesis")[$5 + 3 & & & −5 + ( −3 ) & & & −5 + 3 & & & 5 + ( −3 )$]To add #math.equation(block: false, alt: "5 plus 3 ,")[$5 + 3 ,$] we realize that #math.equation(block: false, alt: "5 plus 3")[$5 + 3$] means the sum of 5 and 3. #figure(table( columns: 2, align: left, inset: 6pt, table.header([We start with 5 positives.], [#figure(figph[Five light blue circles are horizontally arranged, with the number '5' positioned directly below them, illustrating the quantity of five.], alt: "Five light blue circles are horizontally arranged, with the number '5' positioned directly below them, illustrating the quantity of five.", caption: none)]), [And then we add 3 positives.], [#figure(figph[Two groups of light blue circles are shown, with five circles in the first group and three circles in the second, labeled with their respective numbers below.], alt: "Two groups of light blue circles are shown, with five circles in the first group and three circles in the second, labeled with their respective numbers below.", caption: none)], [We now have 8 positives. The sum of 5 and 3 is 8.], [#figure(figph[Eight light blue circles are arranged in a horizontal row, with the text '8 positives' centered beneath them, indicating a count of positive instances represented by the circles.], alt: "Eight light blue circles are arranged in a horizontal row, with the text '8 positives' centered beneath them, indicating a count of positive instances represented by the circles.", caption: none)], )) Now we will add #math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis .")[$−5 + ( −3 ) .$] Watch for similarities to the last example #math.equation(block: false, alt: "5 plus 3 equals 8 .")[$5 + 3 = 8 .$] To add #math.equation(block: false, alt: "−5 plus open parenthesis −3 close parenthesis ,")[$−5 + ( −3 ) ,$] we realize this means the sum of #math.equation(block: false, alt: "−5 and minus 3 .")[$−5 #h(0.2em) "and" #h(0.2em) − 3 .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([We start with 5 negatives.], [#figure(figph[An illustration showing five red circles arranged in a horizontal line, with the number -5 written directly below the center of the line of circles. This represents the concept of negative five.], alt: "An illustration showing five red circles arranged in a horizontal line, with the number -5 written directly below the center of the line of circles. This represents the concept of negative five.", caption: none)]), [And then we add 3 negatives.], [#figure(figph[An illustration displays two groups of red circles, with five circles in the first group labeled -5, and three circles in the second group labeled -3, representing negative numbers.], alt: "An illustration displays two groups of red circles, with five circles in the first group labeled -5, and three circles in the second group labeled -3, representing negative numbers.", caption: none)], [We now have 8 negatives. The sum of −5 and −3 is −8.], [#figure(figph[A horizontal row of eight red circles is displayed, with the text '8 negatives' centered beneath them.], alt: "A horizontal row of eight red circles is displayed, with the text '8 negatives' centered beneath them.", caption: none)], )) In what ways were these first two examples similar? - The first example adds 5 positives and 3 positives—both positives. - The second example adds 5 negatives and 3 negatives—both negatives. In each case we got 8—either 8 positives or 8 negatives. When the signs were the same, the counters were all the same color, and so we added them. #figure(figph[This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text “8 positives.” Centered under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase “8 negatives”. Centered under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.], alt: "This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text “8 positives.” Centered under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase “8 negatives”. Centered under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.", caption: none) #examplebox("Example 8")[][ Add: ⓐ #math.equation(block: false, alt: "1 plus 4")[$1 + 4$] ⓑ #math.equation(block: false, alt: "−1 plus open parenthesis −4 close parenthesis .")[$−1 + ( −4 ) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 1, align: left, inset: 6pt, [#figure(figph[Five light blue circles arranged horizontally, illustrating the sum of 1 and 4, with '1 + 4' and '5' written next to them, representing a basic addition problem.], alt: "Five light blue circles arranged horizontally, illustrating the sum of 1 and 4, with '1 + 4' and '5' written next to them, representing a basic addition problem.", caption: none)], [1 positive plus 4 positives is 5 positives.], )) ⓑ #linebreak() #figure(table( columns: 1, align: left, inset: 6pt, [#figure(figph[Five red circles visually represent the sum of -1 and -4, equaling -5, illustrating the addition of negative numbers.], alt: "Five red circles visually represent the sum of -1 and -4, equaling -5, illustrating the addition of negative numbers.", caption: none)], [1 negative plus 4 negatives is 5 negatives.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: ⓐ #math.equation(block: false, alt: "2 plus 4")[$2 + 4$] ⓑ #math.equation(block: false, alt: "−2 plus open parenthesis −4 close parenthesis .")[$−2 + ( −4 ) .$] #solutionbox[ ⓐ 6 ⓑ #math.equation(block: false, alt: "−6")[$−6$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: ⓐ #math.equation(block: false, alt: "2 plus 5")[$2 + 5$] ⓑ #math.equation(block: false, alt: "−2 plus open parenthesis −5 close parenthesis .")[$−2 + ( −5 ) .$] #solutionbox[ ⓐ 7 ⓑ #math.equation(block: false, alt: "−7")[$−7$] ] ] So what happens when the signs are different? Let’s add #math.equation(block: false, alt: "−5 plus 3 .")[$−5 + 3 .$] We realize this means the sum of #math.equation(block: false, alt: "−5")[$−5$] and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [−5 + 3 means the sum of −5 and 3.]), [We start with 5 negatives.], [#figure(figph[Five red oval shapes arranged horizontally on a white background, resembling a row of pills or beads.], alt: "Five red oval shapes arranged horizontally on a white background, resembling a row of pills or beads.", caption: none)], [And then we add 3 positives.], [#figure(figph[Five red circles are arranged horizontally above three light blue circles, also arranged horizontally, on a white background.], alt: "Five red circles are arranged horizontally above three light blue circles, also arranged horizontally, on a white background.", caption: none)], [We remove any neutral pairs.], [#figure(figph[Three purple ovals, each containing one orange and one light blue circle, are arranged in a row. To the right of these ovals, two standalone orange circles complete the linear arrangement.], alt: "Three purple ovals, each containing one orange and one light blue circle, are arranged in a row. To the right of these ovals, two standalone orange circles complete the linear arrangement.", caption: none)], [We have 2 negatives left.], [#figure(figph[Two red circles are shown above the text '2 negatives' on a white background, suggesting a concept related to negative values or states.], alt: "Two red circles are shown above the text '2 negatives' on a white background, suggesting a concept related to negative values or states.", caption: none)], [The sum of −5 and 3 is −2.], [−5 + 3 = −2], )) Notice that there were more negatives than positives, so the result was negative. Let’s now add the last combination, #math.equation(block: false, alt: "5 plus open parenthesis − 3 close parenthesis .")[$5 + ( "−" 3 ) .$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [5 + (−3) means the sum of 5 and −3.]), [We start with 5 positives.], [#figure(figph[Five light blue, semi-transparent ovals in a neat row on a white background, resembling water droplets or bubbles.], alt: "Five light blue, semi-transparent ovals in a neat row on a white background, resembling water droplets or bubbles.", caption: none)], [And then we add 3 negatives.], [#figure(figph[Eight circles are arranged in two rows on a white background. The top row contains five light blue circles, and the bottom row has three red circles.], alt: "Eight circles are arranged in two rows on a white background. The top row contains five light blue circles, and the bottom row has three red circles.", caption: none)], [We remove any neutral pairs.], [#figure(figph[An illustration featuring three pairs of light blue and orange circles, each pair enclosed in a purple oval, alongside two standalone light blue circles.], alt: "An illustration featuring three pairs of light blue and orange circles, each pair enclosed in a purple oval, alongside two standalone light blue circles.", caption: none)], [We have 2 positives left.], [#figure(figph[Two light blue outlined circles are shown above the text '2 positives', indicating a count of two positive results or items.], alt: "Two light blue outlined circles are shown above the text '2 positives', indicating a count of two positive results or items.", caption: none)], [The sum of 5 and −3 is 2.], [5 + (−3) = 2], )) When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative. #figure(figph[Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “negative 5 plus 3” and below is written “More negatives – the sum is negative.” The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “5 plus negative 3” and below is written “More positives – the sum is positive.”], alt: "Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “negative 5 plus 3” and below is written “More negatives – the sum is negative.” The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “5 plus negative 3” and below is written “More positives – the sum is positive.”", caption: none) #examplebox("Example 9")[][ Add: ⓐ #math.equation(block: false, alt: "−1 plus 5")[$−1 + 5$] ⓑ #math.equation(block: false, alt: "1 plus open parenthesis −5 close parenthesis .")[$1 + ( −5 ) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [−1 + 5]), [], [#figure(figph[A red circle and a blue circle are enclosed by a purple oval, with five more blue circles placed sequentially to the right on a white background.], alt: "A red circle and a blue circle are enclosed by a purple oval, with five more blue circles placed sequentially to the right on a white background.", caption: none)], [There are more positives, so the sum is positive.], [4], )) #linebreak() ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [1 + (−5)]), [], [#figure(figph[A blue circle and a red circle are grouped within a purple oval, while five additional red circles are arranged in a horizontal line next to the group.], alt: "A blue circle and a red circle are grouped within a purple oval, while five additional red circles are arranged in a horizontal line next to the group.", caption: none)], [There are more negatives, so the sum is negative.], [−4], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: ⓐ #math.equation(block: false, alt: "−2 plus 4")[$−2 + 4$] ⓑ #math.equation(block: false, alt: "2 plus open parenthesis −4 close parenthesis .")[$2 + ( −4 ) .$] #solutionbox[ ⓐ 2 ⓑ #math.equation(block: false, alt: "−2")[$−2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Add: ⓐ #math.equation(block: false, alt: "−2 plus 5")[$−2 + 5$] ⓑ #math.equation(block: false, alt: "2 plus open parenthesis −5 close parenthesis .")[$2 + ( −5 ) .$] #solutionbox[ ⓐ 3 ⓑ #math.equation(block: false, alt: "−3")[$−3$] ] ] Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers. When you need to add numbers such as #math.equation(block: false, alt: "37 plus open parenthesis − 53 close parenthesis ,")[$37 + ( "−" 53 ) ,$] you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem? Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be #emph[negative]. How many more red counters would there be? Because #math.equation(block: false, alt: "53 minus 37 equals 16 ,")[$53 − 37 = 16 ,$] there are 16 more red counters. Therefore, the sum of #math.equation(block: false, alt: "37 plus open parenthesis − 53 close parenthesis")[$37 + ( "−" 53 )$] is #math.equation(block: false, alt: "−16 .")[$−16 .$] #math.equation(block: true, alt: "37 plus open parenthesis − 53 close parenthesis equals −16")[$37 + ( "−" 53 ) = −16$]Let’s try another one. We’ll add #math.equation(block: false, alt: "−74 plus open parenthesis − 27 close parenthesis .")[$−74 + ( "−" 27 ) .$] Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is #math.equation(block: false, alt: "−101 .")[$−101 .$] #math.equation(block: true, alt: "−74 plus open parenthesis − 27 close parenthesis equals −101")[$−74 + ( "−" 27 ) = −101$]Let’s look again at the results of adding the different combinations of #math.equation(block: false, alt: "5 , −5")[$5 , −5$] and #math.equation(block: false, alt: "3 , −3 .")[$3 , −3 .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Addition of Positive and Negative Integers] #math.equation(block: true, alt: "5 plus 3, −5 plus open parenthesis −3 close parenthesis; 8, −8; both positive, sum positive, both negative, sum negative")[$5 + 3 & & & −5 + ( −3 ) \ 8 & & & −8 \ "both positive, sum positive" & & & "both negative, sum negative"$]When the signs are the same, the counters would be all the same color, so add them. #math.equation(block: true, alt: "−5 plus 3, 5 plus open parenthesis −3 close parenthesis; −2, 2; different signs, more negatives, sum negative, different signs, more positives, sum positive")[$−5 + 3 & & & 5 + ( −3 ) \ −2 & & & 2 \ "different signs, more negatives, sum negative" & & & "different signs, more positives, sum positive"$]When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left. ] Visualize the model as you simplify the expressions in the following examples. #examplebox("Example 10")[][ Simplify: ⓐ #math.equation(block: false, alt: "19 plus open parenthesis −47 close parenthesis")[$19 + ( −47 )$] ⓑ #math.equation(block: false, alt: "−14 plus open parenthesis −36 close parenthesis .")[$−14 + ( −36 ) .$] #solutionbox[ + ⓐ Since the signs are different, we subtract #math.equation(block: false, alt: "19 from 47 .")[$"19 from 47" "."$] The answer will be negative because there are more negatives than positives. #linebreak() #math.equation(block: false, alt: "19 plus open parenthesis −47 close parenthesis; Add., −28")[$& & & #h(0.3em) 19 + ( −47 ) \ "Add." & & & #h(0.3em) −28$] + ⓑ Since the signs are the same, we add. The answer will be negative because there are only negatives. #linebreak() #math.equation(block: false, alt: "−14 plus open parenthesis −36 close parenthesis; Add., −50")[$& & & −14 + ( −36 ) \ "Add." & & & −50$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "−31 plus open parenthesis −19 close parenthesis")[$−31 + ( −19 )$] ⓑ #math.equation(block: false, alt: "15 plus open parenthesis −32 close parenthesis .")[$15 + ( −32 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−50")[$−50$] ⓑ #math.equation(block: false, alt: "−17")[$−17$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "−42 plus open parenthesis −28 close parenthesis")[$−42 + ( −28 )$] ⓑ #math.equation(block: false, alt: "25 plus open parenthesis −61 close parenthesis .")[$25 + ( −61 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−70")[$−70$] ⓑ #math.equation(block: false, alt: "−36")[$−36$] ] ] The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations! #examplebox("Example 11")[][ Simplify: #math.equation(block: false, alt: "−5 plus 3 open parenthesis −2 plus 7 close parenthesis .")[$−5 + 3 ( −2 + 7 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "−5 plus 3 open parenthesis −2 plus 7 close parenthesis")[$−5 + 3 ( −2 + 7 )$]]), [Simplify inside the parentheses.], [#math.equation(block: false, alt: "−5 plus 3 open parenthesis 5 close parenthesis")[$−5 + 3 ( 5 )$]], [Multiply.], [#math.equation(block: false, alt: "−5 plus 15")[$−5 + 15$]], [Add left to right.], [#math.equation(block: false, alt: "10")[$10$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "−2 plus 5 open parenthesis −4 plus 7 close parenthesis .")[$−2 + 5 ( −4 + 7 ) .$] #solutionbox[ 13 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "−4 plus 2 open parenthesis −3 plus 5 close parenthesis .")[$−4 + 2 ( −3 + 5 ) .$] #solutionbox[ 0 ] ] === Subtract Integers We will continue to use counters to model the 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 #math.equation(block: false, alt: "“ 5")[$"“" 5$] take away #math.equation(block: false, alt: "3. ”")[$3. "”"$] When you use counters, you can think of subtraction the same way! We will model the 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 3, 5 minus open parenthesis −3 close parenthesis")[$5 − 3 & & & −5 − ( −3 ) & & & −5 − 3 & & & 5 − ( −3 )$]To subtract #math.equation(block: false, alt: "5 minus 3 ,")[$5 − 3 ,$] we restate the problem as #math.equation(block: false, alt: "“ 5")[$"“" 5$] take away #math.equation(block: false, alt: "3. ”")[$3. "”"$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([We start with 5 positives.], [#figure(figph[Five light blue, oval-shaped capsules or blobs are arranged horizontally across a white background, suggesting a simple, abstract pattern or representation.], alt: "Five light blue, oval-shaped capsules or blobs are arranged horizontally across a white background, suggesting a simple, abstract pattern or representation.", caption: none)]), [We ‘take away’ 3 positives.], [#figure(figph[Three light blue circles are encircled by a magenta oval with an arrow pointing left, part of a row of five circles.], alt: "Three light blue circles are encircled by a magenta oval with an arrow pointing left, part of a row of five circles.", caption: none)], [We have 2 positives left.], [], [The difference of 5 and 3 is 2.], [2], )) Now we will subtract #math.equation(block: false, alt: "−5 minus open parenthesis −3 close parenthesis .")[$−5 − ( −3 ) .$] Watch for similarities to the last example #math.equation(block: false, alt: "5 minus 3 equals 2 .")[$5 − 3 = 2 .$] To subtract #math.equation(block: false, alt: "−5 minus open parenthesis −3 close parenthesis ,")[$−5 − ( −3 ) ,$] we restate this as #math.equation(block: false, alt: "“ –5")[$"“" –5$] take away #math.equation(block: false, alt: "–3 ”")[$–3 "”"$] #figure(table( columns: 2, align: left, inset: 6pt, table.header([We start with 5 negatives.], [#figure(figph[A row of five identical reddish-orange oval shapes, possibly representing beads or segments, aligned horizontally against a white background.], alt: "A row of five identical reddish-orange oval shapes, possibly representing beads or segments, aligned horizontally against a white background.", caption: none)]), [We ‘take away’ 3 negatives.], [#figure(figph[A row of five orange circles, with the first three grouped by a purple oval and an arrow indicating this grouping. This illustrates counting or grouping three items from a set of five.], alt: "A row of five orange circles, with the first three grouped by a purple oval and an arrow indicating this grouping. This illustrates counting or grouping three items from a set of five.", caption: none)], [We have 2 negatives left.], [], [The difference of −5 and −3 is −2.], [−2], )) Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives. In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives. Each example used counters of only one color, and the “take away” model of subtraction was easy to apply. #figure(figph[Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation “5 minus 3 equals 2.” The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation “negative 5, minus, negative 3, equals negative 2.”], alt: "Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation “5 minus 3 equals 2.” The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation “negative 5, minus, negative 3, equals negative 2.”", caption: none) #examplebox("Example 12")[][ Subtract: ⓐ #math.equation(block: false, alt: "7 minus 5")[$7 − 5$] ⓑ #math.equation(block: false, alt: "−7 minus open parenthesis −5 close parenthesis .")[$−7 − ( −5 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() Take 5 positive from 7 positives and get 2 positives.], [#math.equation(block: false, alt: "7 minus 5; 2")[$7 − 5 \ 2$]]), [ⓑ #linebreak() Take 5 negatives from 7 negatives and get 2 negatives.], [#math.equation(block: false, alt: "−7 minus open parenthesis −5 close parenthesis; −2")[$−7 − ( −5 ) \ −2$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: ⓐ #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] ⓑ #math.equation(block: false, alt: "−6 minus open parenthesis −4 close parenthesis .")[$−6 − ( −4 ) .$] #solutionbox[ ⓐ 2 ⓑ #math.equation(block: false, alt: "−2")[$−2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: ⓐ #math.equation(block: false, alt: "7 minus 4")[$7 − 4$] ⓑ #math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis .")[$−7 − ( −4 ) .$] #solutionbox[ ⓐ 3 ⓑ #math.equation(block: false, alt: "−3")[$−3$] ] ] What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some #strong[neutral pairs]. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different. - To subtract #math.equation(block: false, alt: "−5 minus 3 ,")[$−5 − 3 ,$] we restate it as #math.equation(block: false, alt: "−5")[$−5$] take away 3. We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away. Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [−5 − 3 means −5 take away 3.]), [We start with 5 negatives.], [#figure(figph[Five red circles representing the number -5, illustrating a negative integer concept.], alt: "Five red circles representing the number -5, illustrating a negative integer concept.", caption: none)], [We now add the neutrals needed to get 3 positives.], [#figure(figph[Seven red circles are arranged in a row, with a group of three blue circles directly beneath the last three red circles on a white background.], alt: "Seven red circles are arranged in a row, with a group of three blue circles directly beneath the last three red circles on a white background.", caption: none)], [We remove the 3 positives.], [#figure(figph[An image shows seven red circles in two sets (four then three). Three light blue circles are grouped below the second set of red circles, enclosed by a purple oval and an arrow pointing left.], alt: "An image shows seven red circles in two sets (four then three). Three light blue circles are grouped below the second set of red circles, enclosed by a purple oval and an arrow pointing left.", caption: none)], [We are left with 8 negatives.], [#figure(figph[Eight red circles are arranged horizontally in a row, with the text '8 negatives' written directly below them, indicating a visual representation of eight negative units.], alt: "Eight red circles are arranged horizontally in a row, with the text '8 negatives' written directly below them, indicating a visual representation of eight negative units.", caption: none)], [The difference of −5 and 3 is −8.], [−5 − 3 = −8], )) And now, the fourth case, #math.equation(block: false, alt: "5 minus open parenthesis −3 close parenthesis .")[$5 − ( −3 ) .$] We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [5 − (−3) means 5 take away −3.]), [We start with 5 positives.], [#figure(figph[Five identical light blue circles are arranged in a horizontal line against a plain white background, appearing as simple, evenly spaced graphic elements.], alt: "Five identical light blue circles are arranged in a horizontal line against a plain white background, appearing as simple, evenly spaced graphic elements.", caption: none)], [We now add the needed neutrals pairs.], [#figure(figph[Six light blue circles are arranged in the top row, with three red circles positioned below them.], alt: "Six light blue circles are arranged in the top row, with three red circles positioned below them.", caption: none)], [We remove the 3 negatives.], [#figure(figph[An illustration showing a row of light blue circles with three red circles beneath them, encircled by a purple oval with a curved arrow pointing left.], alt: "An illustration showing a row of light blue circles with three red circles beneath them, encircled by a purple oval with a curved arrow pointing left.", caption: none)], [We are left with 8 positives.], [#figure(figph[Eight light blue circles are arranged in a horizontal row, with the text '8 positives' centered below them.], alt: "Eight light blue circles are arranged in a horizontal row, with the text '8 positives' centered below them.", caption: none)], [The difference of 5 and −3 is 8.], [5 − (−3) = 8], )) #examplebox("Example 13")[][ Subtract: ⓐ #math.equation(block: false, alt: "−3 minus 1")[$−3 − 1$] ⓑ #math.equation(block: false, alt: "3 minus open parenthesis −1 close parenthesis .")[$3 − ( −1 ) .$] #solutionbox[ ⓐ #figure(table( columns: 3, align: left, inset: 6pt, [Take 1 positive from the one added neutral pair.], [#figure(figph[A row of five pale orange circles with red outlines on a white background, with a gap between the fourth and fifth circles.], alt: "A row of five pale orange circles with red outlines on a white background, with a gap between the fourth and fifth circles.", caption: none) #linebreak() #figure(figph[A light blue circle is encircled by a purple elliptical shape with an arrow indicating a clockwise rotation, all set against a white background.], alt: "A light blue circle is encircled by a purple elliptical shape with an arrow indicating a clockwise rotation, all set against a white background.", caption: none)], [−3 − 1 #linebreak() #linebreak() −4], )) ⓑ #figure(table( columns: 3, align: left, inset: 6pt, [Take 1 negative from the one added neutral pair.], [#figure(figph[A row of four light blue circles with a thin, darker blue outline, set against a plain white background. The circles are evenly spaced and identical in appearance.], alt: "A row of four light blue circles with a thin, darker blue outline, set against a plain white background. The circles are evenly spaced and identical in appearance.", caption: none) #linebreak() #figure(figph[Red circle with orange fill and a purple elliptical arrow around it, pointing clockwise.], alt: "Red circle with orange fill and a purple elliptical arrow around it, pointing clockwise.", caption: none)], [3 − (−1) #linebreak() #linebreak() 4], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: ⓐ #math.equation(block: false, alt: "−6 minus 4")[$−6 − 4$] ⓑ #math.equation(block: false, alt: "6 minus open parenthesis −4 close parenthesis .")[$6 − ( −4 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−10")[$−10$] ⓑ 10 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: ⓐ #math.equation(block: false, alt: "−7 minus 4")[$−7 − 4$] ⓑ #math.equation(block: false, alt: "7 minus open parenthesis −4 close parenthesis .")[$7 − ( −4 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−11")[$−11$] ⓑ 11 ] ] Have you noticed that #emph[subtraction of signed numbers can be done by adding the opposite]? In , #math.equation(block: false, alt: "−3 minus 1")[$−3 − 1$] is the same as #math.equation(block: false, alt: "−3 plus open parenthesis −1 close parenthesis")[$−3 + ( −1 )$] and #math.equation(block: false, alt: "3 minus open parenthesis −1 close parenthesis")[$3 − ( −1 )$] is the same as #math.equation(block: false, alt: "3 plus 1 .")[$3 + 1 .$] You will often see this 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")[$a − b = a + ( "−" b )$]Subtracting a number is the same as adding its opposite. ] Look at these two examples. #figure(figph[Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as “6 minus 4” and below as “2.” The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as “6 plus, open parenthesis, negative 4, close parenthesis” and below as “2.”], alt: "Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as “6 minus 4” and below as “2.” The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as “6 plus, open parenthesis, negative 4, close parenthesis” and below as “2.”", caption: none) #math.equation(block: true, alt: "6 minus 4 gives the same answer as 6 plus open parenthesis −4 close parenthesis .")[$6 − 4 #h(0.2em) "gives the same answer as" #h(0.2em) 6 + ( −4 ) .$]Of course, when you have a subtraction problem that has only positive numbers, like #math.equation(block: false, alt: "6 minus 4 ,")[$6 − 4 ,$] you just do the subtraction. You already knew how to subtract #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] long ago. But #emph[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 you are subtracting negative numbers. Make sure that you understand how #math.equation(block: false, alt: "6 minus 4")[$6 − 4$] and #math.equation(block: false, alt: "6 plus open parenthesis −4 close parenthesis")[$6 + ( −4 )$] give the same results! #examplebox("Example 14")[][ Simplify: ⓐ #math.equation(block: false, alt: "13 minus 8")[$13 − 8$] and #math.equation(block: false, alt: "13 plus open parenthesis −8 close parenthesis")[$13 + ( −8 )$] ⓑ #math.equation(block: false, alt: "−17 minus 9")[$−17 − 9$] and #math.equation(block: false, alt: "−17 plus open parenthesis −9 close parenthesis .")[$−17 + ( −9 ) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([ⓐ #linebreak() Subtract.], [#math.equation(block: false, alt: "13 minus 8; 5")[$13 − 8 \ 5$]], [#math.equation(block: false, alt: "13 plus open parenthesis −8 close parenthesis; 5")[$13 + ( −8 ) \ 5$]]), [ⓑ #linebreak() Subtract.], [#math.equation(block: false, alt: "−17 minus 9; −26")[$−17 − 9 \ −26$]], [#math.equation(block: false, alt: "−17 plus open parenthesis −9 close parenthesis; −26")[$−17 + ( −9 ) \ −26$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "21 minus 13")[$21 − 13$] and #math.equation(block: false, alt: "21 plus open parenthesis −13 close parenthesis")[$21 + ( −13 )$] ⓑ #math.equation(block: false, alt: "−11 minus 7")[$−11 − 7$] and #math.equation(block: false, alt: "−11 plus open parenthesis −7 close parenthesis .")[$−11 + ( −7 ) .$] #solutionbox[ ⓐ 8 ⓑ −18 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "15 minus 7")[$15 − 7$] and #math.equation(block: false, alt: "15 plus open parenthesis −7 close parenthesis")[$15 + ( −7 )$] ⓑ #math.equation(block: false, alt: "−14 minus 8")[$−14 − 8$] and #math.equation(block: false, alt: "−14 plus open parenthesis −8 close parenthesis .")[$−14 + ( −8 ) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "8")[$8$] ⓑ #math.equation(block: false, alt: "−22")[$−22$] ] ] Look at what happens when we subtract a negative. #figure(figph[This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.], alt: "This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.", caption: none) #math.equation(block: true, alt: "8 minus open parenthesis −5 close parenthesis gives the same answer as 8 plus 5")[$8 − ( −5 ) #h(0.2em) "gives the same answer as" #h(0.2em) 8 + 5$]#emph[Subtracting a negative number is like adding a positive!] You will often see this written as #math.equation(block: false, alt: "a minus open parenthesis − b close parenthesis equals a plus b .")[$a − ( "−" b ) = a + b .$] Does that work for other numbers, too? Let’s do the following example and see. #examplebox("Example 15")[][ Simplify: ⓐ #math.equation(block: false, alt: "9 minus open parenthesis −15 close parenthesis")[$9 − ( −15 )$] and #math.equation(block: false, alt: "9 plus 15")[$9 + 15$] ⓑ #math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis")[$−7 − ( −4 )$] and #math.equation(block: false, alt: "−7 plus 4 .")[$−7 + 4 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([ⓐ #linebreak() Subtract.], [#math.equation(block: false, alt: "9 minus open parenthesis −15 close parenthesis; 24")[$9 − ( −15 ) \ 24$]], [#math.equation(block: false, alt: "9 plus 15; 24")[$9 + 15 \ 24$]]), [ⓑ #linebreak() Subtract.], [#math.equation(block: false, alt: "−7 minus open parenthesis −4 close parenthesis; −3")[$−7 − ( −4 ) \ −3$]], [#math.equation(block: false, alt: "−7 plus 4; −3")[$−7 + 4 \ −3$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "6 minus open parenthesis −13 close parenthesis")[$6 − ( −13 )$] and #math.equation(block: false, alt: "6 plus 13")[$6 + 13$] ⓑ #math.equation(block: false, alt: "−5 minus open parenthesis −1 close parenthesis")[$−5 − ( −1 )$] and #math.equation(block: false, alt: "−5 plus 1 .")[$−5 + 1 .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "19")[$19$] ⓑ #math.equation(block: false, alt: "−4")[$−4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "4 minus open parenthesis −19 close parenthesis")[$4 − ( −19 )$] and #math.equation(block: false, alt: "4 plus 19")[$4 + 19$] ⓑ #math.equation(block: false, alt: "−4 minus open parenthesis −7 close parenthesis")[$−4 − ( −7 )$] and #math.equation(block: false, alt: "−4 plus 7 .")[$−4 + 7 .$] #solutionbox[ ⓐ 23 ⓑ 3 ] ] Let’s look again at the results of subtracting the different combinations of #math.equation(block: false, alt: "5 , −5")[$5 , −5$] and #math.equation(block: false, alt: "3 , −3 .")[$3 , −3 .$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Subtraction of Integers] #math.equation(block: true, alt: "5 minus 3, −5 minus open parenthesis −3 close parenthesis; 2, −2; 5 positives take away 3 positives, 5 negatives take away 3 negatives; 2 positives, 2 negatives")[$5 − 3 & & & −5 − ( −3 ) \ 2 & & & −2 \ 5 #h(0.2em) "positives take away" #h(0.2em) 3 #h(0.2em) "positives" & & & 5 #h(0.2em) "negatives take away" #h(0.2em) 3 #h(0.2em) "negatives" \ 2 #h(0.2em) "positives" & & & 2 #h(0.2em) "negatives"$]When there would be enough counters of the color to take away, subtract. #math.equation(block: true, alt: "−5 minus 3, 5 minus open parenthesis −3 close parenthesis; −8, 8; 5 negatives, want to take away 3 positives, 5 positives, want to take away 3 negatives; need neutral pairs, need neutral pairs")[$−5 − 3 & & & 5 − ( −3 ) \ −8 & & & 8 \ 5 #h(0.2em) "negatives, want to take away" #h(0.2em) 3 #h(0.2em) "positives" & & & 5 #h(0.2em) "positives, want to take away" #h(0.2em) 3 #h(0.2em) "negatives" \ "need neutral pairs" & & & "need neutral pairs"$]When there would be not enough counters of the color to take away, add. ] What happens when there are more than three integers? We just use the order of operations as usual. #examplebox("Example 16")[][ Simplify: #math.equation(block: false, alt: "7 minus open parenthesis −4 minus 3 close parenthesis minus 9 .")[$7 − ( −4 − 3 ) − 9 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "7 minus open parenthesis −4 minus 3 close parenthesis minus 9")[$7 − ( −4 − 3 ) − 9$]]), [Simplify inside the parentheses first.], [#math.equation(block: false, alt: "7 minus open parenthesis −7 close parenthesis minus 9")[$7 − ( −7 ) − 9$]], [Subtract left to right.], [#math.equation(block: false, alt: "14 minus 9")[$14 − 9$]], [Subtract.], [#math.equation(block: false, alt: "5")[$5$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "8 minus open parenthesis −3 minus 1 close parenthesis minus 9")[$8 − ( −3 − 1 ) − 9$]. #solutionbox[ 3 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "12 minus open parenthesis −9 minus 6 close parenthesis minus 14")[$12 − ( −9 − 6 ) − 14$]. #solutionbox[ 13 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications. - #link("https://openstax.org/l/11AddColorChip")[Add Colored Chip] - #link("https://openstax.org/l/11SubtrColorChp")[Subtract Colored Chip] ] === Key Concepts - #strong[Addition of Positive and Negative Integers] #linebreak() #math.equation(block: false, alt: "5 plus 3, minus 5 plus open parenthesis −3 close parenthesis; 8, −8; both positive,, both negative,; sum positive, sum negative; −5 plus 3, 5 plus open parenthesis −3 close parenthesis; −2, 2; different signs,, different signs,; more negatives, more positives; sum negative, sum positive")[$5 + 3 & & & #h(2.5em) − 5 + ( −3 ) \ #h(0.95em) 8 & & & #h(3.45em) −8 \ "both positive," & & & #h(2.5em) "both negative," \ "sum positive" & & & #h(2.5em) "sum negative" \ \ \ −5 + 3 & & & #h(2.5em) 5 + ( −3 ) \ #h(0.95em) −2 & & & #h(3.45em) 2 \ "different signs," & & & #h(2.5em) "different signs," \ "more negatives" & & & #h(2.5em) "more positives" \ "sum negative" & & & #h(2.5em) "sum positive"$] - #strong[Property of Absolute Value]: #math.equation(block: false, alt: "| n | greater than or equal to 0")[$| n | ≥ 0$] for all numbers. Absolute values are always greater than or equal to zero! - #strong[Subtraction of Integers] #linebreak() #math.equation(block: false, alt: "5 minus 3, −5 minus open parenthesis −3 close parenthesis; 2, −2; 5 positives, 5 negatives; take away 3 positives, take away 3 negatives; 2 positives, 2 negatives; −5 minus 3, 5 minus open parenthesis −3 close parenthesis; −8, 8; 5 negatives, want to, 5 positives, want to; subtract 3 positives, subtract 3 negatives; need neutral pairs, need neutral pairs")[$5 − 3 & & & −5 − ( −3 ) \ #h(0.95em) 2 & & & #h(0.95em) −2 \ 5 #h(0.2em) "positives" & & & 5 #h(0.2em) "negatives" \ "take away" #h(0.2em) 3 #h(0.2em) "positives" & & & "take away" #h(0.2em) 3 #h(0.2em) "negatives" \ "2 positives" & & & "2 negatives" \ \ \ −5 − 3 & & & 5 − ( −3 ) \ #h(0.95em) −8 & & & #h(0.95em) 8 \ 5 #h(0.2em) "negatives, want to" & & & 5 #h(0.2em) "positives, want to" \ "subtract" #h(0.2em) 3 #h(0.2em) "positives" & & & "subtract" #h(0.2em) 3 #h(0.2em) "negatives" \ "need neutral pairs" & & & "need neutral pairs"$] - #strong[Subtraction Property:] Subtracting a number is the same as adding its opposite. ==== Practice Makes Perfect #strong[Use Negatives and Opposites of Integers] In the following exercises, order each of the following pairs of numbers, using \< or \>. ⓐ #math.equation(block: false, alt: "9 ___ 4")[$9 \_\_\_ 4$] #linebreak() ⓑ #math.equation(block: false, alt: "−3 ___ 6")[$−3 \_\_\_ 6$] #linebreak() ⓒ #math.equation(block: false, alt: "−8 ___ −2")[$−8 \_\_\_ −2$] #linebreak() ⓓ #math.equation(block: false, alt: "1 ___ −10")[$1 \_\_\_ −10$] #solutionbox[ ⓐ \> ⓑ \< ⓒ \< ⓓ \> ] ⓐ #math.equation(block: false, alt: "−7 ___ 3")[$−7 \_\_\_ 3$] #linebreak() ⓑ #math.equation(block: false, alt: "−10 ___ −5")[$−10 \_\_\_ −5$] #linebreak() ⓒ #math.equation(block: false, alt: "2 ___ −6")[$2 \_\_\_ −6$] #linebreak() ⓓ #math.equation(block: false, alt: "8 ___ 9")[$8 \_\_\_ 9$] In the following exercises, find the opposite of each number. ⓐ 2 ⓑ #math.equation(block: false, alt: "−6")[$−6$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ 6 ] ⓐ 9ⓑ #math.equation(block: false, alt: "−4")[$−4$] In the following exercises, simplify. #math.equation(block: true, alt: "− open parenthesis −4 close parenthesis")[$"−" ( −4 )$] #solutionbox[ 4 ] #math.equation(block: true, alt: "− open parenthesis −8 close parenthesis")[$"−" ( −8 )$] #math.equation(block: true, alt: "− open parenthesis −15 close parenthesis")[$"−" ( −15 )$] #solutionbox[ 15 ] #math.equation(block: true, alt: "− open parenthesis −11 close parenthesis")[$"−" ( −11 )$] In the following exercises, evaluate. #math.equation(block: false, alt: "− c")[$"−" c$] whenⓐ #math.equation(block: false, alt: "c equals 12")[$c = 12$]ⓑ #math.equation(block: false, alt: "c equals −12")[$c = −12$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−12")[$−12$] ⓑ 12 ] #math.equation(block: false, alt: "− d")[$"−" d$] when #linebreak() ⓐ #math.equation(block: false, alt: "d equals 21")[$d = 21$] #linebreak() ⓑ #math.equation(block: false, alt: "d equals −21")[$d = −21$] #strong[Simplify Expressions with Absolute Value] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "| −32 |")[$| −32 |$]ⓑ #math.equation(block: false, alt: "| 0 |")[$| 0 |$]ⓒ #math.equation(block: false, alt: "| 16 |")[$| 16 |$] #solutionbox[ ⓐ 32 ⓑ 0 ⓒ 16 ] ⓐ #math.equation(block: false, alt: "| 0 |")[$| 0 |$] #linebreak() ⓑ #math.equation(block: false, alt: "| −40 |")[$| −40 |$]ⓒ #math.equation(block: false, alt: "| 22 |")[$| 22 |$] In the following exercises, fill in \<, \>, or #math.equation(block: false, alt: "equals")[$=$] for each of the following pairs of numbers. ⓐ #math.equation(block: false, alt: "−6 ___ | −6 |")[$−6 \_\_\_ | −6 |$]ⓑ #math.equation(block: false, alt: "− | −3 | ___ −3")[$"−" | −3 | \_\_\_ −3$] #solutionbox[ ⓐ \< ⓑ #math.equation(block: false, alt: "equals")[$=$] ] ⓐ #math.equation(block: false, alt: "| −5 | ___ − | −5 |")[$| −5 | \_\_\_ "−" | −5 |$]ⓑ #math.equation(block: false, alt: "9 ___ − | −9 |")[$9 \_\_\_ "−" | −9 |$] In the following exercises, simplify. #math.equation(block: true, alt: "− open parenthesis −5 close parenthesis and − | −5 |")[$"−" ( −5 ) #h(0.2em) "and" #h(0.2em) "−" | −5 |$] #solutionbox[ #math.equation(block: true, alt: "5 , −5")[$5 , −5$] ] #math.equation(block: true, alt: "− | −9 | and − open parenthesis −9 close parenthesis")[$"−" | −9 | #h(0.2em) "and" #h(0.2em) "−" ( −9 )$] #math.equation(block: true, alt: "8 | −7 |")[$8 | −7 |$] #solutionbox[ 56 ] #math.equation(block: true, alt: "5 | −5 |")[$5 | −5 |$] #math.equation(block: true, alt: "| 15 minus 7 | minus | 14 minus 6 |")[$| 15 − 7 | − | 14 − 6 |$] #solutionbox[ 0 ] #math.equation(block: true, alt: "| 17 minus 8 | minus | 13 minus 4 |")[$| 17 − 8 | − | 13 − 4 |$] #math.equation(block: true, alt: "18 minus | 2 open parenthesis 8 minus 3 close parenthesis |")[$18 − | 2 ( 8 − 3 ) |$] #solutionbox[ 8 ] #math.equation(block: true, alt: "18 minus | 3 open parenthesis 8 minus 5 close parenthesis |")[$18 − | 3 ( 8 − 5 ) |$] In the following exercises, evaluate. ⓐ #math.equation(block: false, alt: "− | p | when p equals 19")[$"−" | p | #h(0.2em) "when" #h(0.2em) p = 19$] #linebreak() ⓑ #math.equation(block: false, alt: "− | q | when q equals −33")[$"−" | q | #h(0.2em) "when" #h(0.2em) q = −33$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−19")[$−19$] ⓑ #math.equation(block: false, alt: "−33")[$−33$] ] ⓐ #math.equation(block: false, alt: "− | a | when a equals 60")[$"−" | a | #h(0.2em) "when" #h(0.2em) a = 60$] #linebreak() ⓑ #math.equation(block: false, alt: "− | b | when b equals −12")[$"−" | b | #h(0.2em) "when" #h(0.2em) b = −12$] #strong[Add Integers] In the following exercises, simplify each expression. #math.equation(block: true, alt: "−21 plus open parenthesis −59 close parenthesis")[$−21 + ( −59 )$] #solutionbox[ #math.equation(block: true, alt: "−80")[$−80$] ] #math.equation(block: true, alt: "−35 plus open parenthesis −47 close parenthesis")[$−35 + ( −47 )$] #math.equation(block: true, alt: "48 plus open parenthesis −16 close parenthesis")[$48 + ( −16 )$] #solutionbox[ 32 ] #math.equation(block: true, alt: "34 plus open parenthesis −19 close parenthesis")[$34 + ( −19 )$] #math.equation(block: true, alt: "−14 plus open parenthesis −12 close parenthesis plus 4")[$−14 + ( −12 ) + 4$] #solutionbox[ #math.equation(block: true, alt: "−22")[$−22$] ] #math.equation(block: true, alt: "−17 plus open parenthesis −18 close parenthesis plus 6")[$−17 + ( −18 ) + 6$] #math.equation(block: true, alt: "135 plus open parenthesis −110 close parenthesis plus 83")[$135 + ( −110 ) + 83$] #solutionbox[ 108 ] #math.equation(block: true, alt: "−38 plus 27 plus open parenthesis −8 close parenthesis plus 12")[$−38 + 27 + ( −8 ) + 12$] #math.equation(block: true, alt: "19 plus 2 open parenthesis −3 plus 8 close parenthesis")[$19 + 2 ( −3 + 8 )$] #solutionbox[ 29 ] #math.equation(block: true, alt: "24 plus 3 open parenthesis −5 plus 9 close parenthesis")[$24 + 3 ( −5 + 9 )$] #strong[Subtract Integers] In the following exercises, simplify. #math.equation(block: true, alt: "8 minus 2")[$8 − 2$] #solutionbox[ 6 ] #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[ #math.equation(block: true, alt: "−9")[$−9$] ] #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[ 12 ] #math.equation(block: true, alt: "7 minus open parenthesis −3 close parenthesis")[$7 − ( −3 )$] ⓐ #math.equation(block: false, alt: "44 minus 28")[$44 − 28$]ⓑ #math.equation(block: false, alt: "44 plus open parenthesis −28 close parenthesis")[$44 + ( −28 )$] #solutionbox[ ⓐ 16 ⓑ 16 ] ⓐ #math.equation(block: false, alt: "35 minus 16")[$35 − 16$]ⓑ #math.equation(block: false, alt: "35 plus open parenthesis −16 close parenthesis")[$35 + ( −16 )$] ⓐ #math.equation(block: false, alt: "27 minus open parenthesis −18 close parenthesis")[$27 − ( −18 )$]ⓑ #math.equation(block: false, alt: "27 plus 18")[$27 + 18$] #solutionbox[ ⓐ 45 ⓑ 45 ] ⓐ #math.equation(block: false, alt: "46 minus open parenthesis −37 close parenthesis")[$46 − ( −37 )$]ⓑ #math.equation(block: false, alt: "46 plus 37")[$46 + 37$] 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: "48 minus 87")[$48 − 87$] #solutionbox[ #math.equation(block: true, alt: "−39")[$−39$] ] #math.equation(block: true, alt: "45 minus 69")[$45 − 69$] #math.equation(block: true, alt: "−17 minus 42")[$−17 − 42$] #solutionbox[ #math.equation(block: true, alt: "−59")[$−59$] ] #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[ #math.equation(block: true, alt: "−51")[$−51$] ] #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[ #math.equation(block: true, alt: "−2")[$−2$] ] #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[ #math.equation(block: true, alt: "−2")[$−2$] ] #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: "64 plus open parenthesis −17 close parenthesis minus 9")[$64 + ( −17 ) − 9$] #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: "− 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: "− 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 [ 10 minus open parenthesis 3 minus 12 close parenthesis ]")[$25 − [ 10 − ( 3 − 12 ) ]$] #solutionbox[ 6 ] #math.equation(block: true, alt: "32 minus [ 5 minus open parenthesis 15 minus 20 close parenthesis ]")[$32 − [ 5 − ( 15 − 20 ) ]$] #math.equation(block: true, alt: "6 times 3 minus 4 times 3 minus 7 times 2")[$6 ⋅ 3 − 4 ⋅ 3 − 7 ⋅ 2$] #solutionbox[ #math.equation(block: true, alt: "−8")[$−8$] ] #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[ #math.equation(block: true, alt: "−11")[$−11$] ] #math.equation(block: true, alt: "6 squared minus 7 squared")[$6^(2) − 7^(2)$] ==== Everyday Math #strong[Elevation] The highest elevation in the United States is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level. Use integers to write the elevation of: ⓐ Mount McKinley. ⓑ Death Valley. #solutionbox[ ⓐ 20,320 feet ⓑ −282 feet ] #strong[Extreme temperatures] The highest recorded temperature on Earth was #math.equation(block: false, alt: "57 °")[$57 "°"$] Celsius. The lowest recorded temperature was #math.equation(block: false, alt: "90 ° below 0 °")[$90 "°" #h(0.2em) "below" #h(0.2em) 0 "°"$] Celsius. Use integers to write the: + ⓐ highest recorded temperature. + ⓑ lowest recorded temperature. #strong[State budgets] In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of \$540 million. That same month, Texas estimated it would have a budget deficit of \$27 billion. Use integers to write the budget of: ⓐ Pennsylvania. #linebreak() ⓑ Texas. #solutionbox[ ⓐ \$540 million ⓑ #math.equation(block: false, alt: "− $27")[$"−" \$27$] billion ] #strong[College enrollments] Across the United States, community college enrollment grew by 1,400,000 students from Fall 2007 to Fall 2010. In California, community college enrollment declined by 110,171 students from Fall 2009 to Fall 2010. Use integers to write the change in enrollment: + ⓐ in the U.S. from Fall 2007 to Fall 2010. + ⓑ in California from Fall 2009 to Fall 2010. #strong[Stock Market] The week of September 15, 2008 was one of the most volatile weeks ever for the US stock market. The closing numbers of the Dow Jones Industrial Average each day were: #figure(table( columns: 2, align: left, inset: 6pt, table.header([Monday], [#math.equation(block: false, alt: "−504")[$−504$]]), [Tuesday], [#math.equation(block: false, alt: "plus 142")[$+ 142$]], [Wednesday], [#math.equation(block: false, alt: "−449")[$−449$]], [Thursday], [#math.equation(block: false, alt: "plus 410")[$+ 410$]], [Friday], [#math.equation(block: false, alt: "plus 369")[$+ 369$]], )) What was the overall change for the week? Was it positive or negative? #solutionbox[ 32, negative ] #strong[Stock Market] During the week of June 22, 2009, the closing numbers of the Dow Jones Industrial Average each day were: #figure(table( columns: 2, align: left, inset: 6pt, table.header([Monday], [#math.equation(block: false, alt: "−201")[$−201$]]), [Tuesday], [#math.equation(block: false, alt: "−16")[$−16$]], [Wednesday], [#math.equation(block: false, alt: "−23")[$−23$]], [Thursday], [#math.equation(block: false, alt: "plus 172")[$+ 172$]], [Friday], [#math.equation(block: false, alt: "−34")[$−34$]], )) What was the overall change for the week? Was it positive or negative? ==== Writing Exercises Give an example of a negative number from your life experience. #solutionbox[ Answers may vary ] What are the three uses of the #math.equation(block: false, alt: "“ minus ”")[$"“" − "”"$] sign in algebra? Explain how they differ. Explain why the sum of #math.equation(block: false, alt: "−8")[$−8$] and 2 is negative, but the sum of 8 and #math.equation(block: false, alt: "−2")[$−2$] is positive. #solutionbox[ Answers may vary ] Give an example from your life experience of adding two negative numbers. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A table is shown with four columns and five rows. The column titles, from left to right, are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!” The first column includes the phrases “use negatives and opposites of integers.”, “Simplify: expressions with absolute value.”, “add integers.” and “subtract integers.”], alt: "A table is shown with four columns and five rows. The column titles, from left to right, are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!” The first column includes the phrases “use negatives and opposites of integers.”, “Simplify: expressions with absolute value.”, “add integers.” and “subtract integers.”", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?