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== 1.4#h(0.6em)Multiply and Divide 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].
]


=== Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show #strong[multiplication] of #strong[integers]. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that #math.equation(block: false, alt: "a times b")[$a · b$] means add #emph[a], #emph[b] times.

#figure(figph[Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”], alt: "Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”", caption: none)

The next two examples are more interesting.

What does it mean to multiply 5 by #math.equation(block: false, alt: "−3 ?")[$−3 ?$] It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

#figure(figph[This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.], alt: "This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.", caption: none)

In summary:

#math.equation(block: true, alt: "5 times 3, equals, 15, −5 open parenthesis 3 close parenthesis, equals, −15; 5 open parenthesis −3 close parenthesis, equals, −15, open parenthesis −5 close parenthesis open parenthesis −3 close parenthesis, equals, 15")[$5 · 3 & = & 15 &  &  & −5 ( 3 ) & = & −15 \ 5 ( −3 ) & = & −15 &  &  & ( −5 ) ( −3 ) & = & 15$]Notice that for multiplication of two signed numbers, when the:

- signs are the #emph[same], the product is #emph[positive].
- signs are #emph[different], the product is #emph[negative].

We’ll put this all together in the chart below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication of Signed Numbers]

For multiplication of two signed numbers:

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Same signs], [Product], [Example]),
  [Two positives #linebreak() Two negatives], [Positive #linebreak() Positive], [#math.equation(block: false, alt: "7 times 4, equals, 28; −8 open parenthesis −6 close parenthesis, equals, 48")[$7 · 4 & = & 28 \ −8 ( −6 ) & = & 48$]],
))

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Different signs], [Product], [Example]),
  [Positive · negative #linebreak() Negative · positive], [Negative #linebreak() Negative], [#math.equation(block: false, alt: "7 open parenthesis −9 close parenthesis, equals, −63; −5 times 10, equals, −50")[$7 ( −9 ) & = & −63 \ −5 · 10 & = & −50$]],
))
]

#examplebox("Example 1")[][
Multiply: ⓐ #math.equation(block: false, alt: "−9 times 3")[$−9 · 3$] ⓑ #math.equation(block: false, alt: "−2 open parenthesis −5 close parenthesis")[$−2 ( −5 )$] ⓒ #math.equation(block: false, alt: "4 open parenthesis −8 close parenthesis")[$4 ( −8 )$] ⓓ #math.equation(block: false, alt: "7 times 6 .")[$7 · 6 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ  #linebreak() 
    Multiply, noting that the signs are different so the product is negative.], [#math.equation(block: false, alt: "−9 times 3; −27")[$−9 · 3 \ −27$]]),
  [ⓑ  #linebreak() 
    Multiply, noting that the signs are the same so the product is positive.], [#math.equation(block: false, alt: "−2 open parenthesis −5 close parenthesis; 10")[$−2 ( −5 ) \ 10$]],
  [ⓒ  #linebreak() 
    Multiply, with different signs.], [#math.equation(block: false, alt: "4 open parenthesis −8 close parenthesis; −32")[$4 ( −8 ) \ −32$]],
  [ⓓ  #linebreak() Multiply, with same signs.], [#math.equation(block: false, alt: "7 times 6; 42")[$7 · 6 \ 42$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: ⓐ #math.equation(block: false, alt: "−6 times 8")[$−6 · 8$] ⓑ #math.equation(block: false, alt: "−4 open parenthesis −7 close parenthesis")[$−4 ( −7 )$] ⓒ #math.equation(block: false, alt: "9 open parenthesis −7 close parenthesis")[$9 ( −7 )$] ⓓ #math.equation(block: false, alt: "5 times 12 .")[$5 · 12 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−48")[$−48$] ⓑ 28 ⓒ #math.equation(block: false, alt: "−63")[$−63$] ⓓ 60
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: ⓐ #math.equation(block: false, alt: "−8 times 7")[$−8 · 7$] ⓑ #math.equation(block: false, alt: "−6 open parenthesis −9 close parenthesis")[$−6 ( −9 )$] ⓒ #math.equation(block: false, alt: "7 open parenthesis −4 close parenthesis")[$7 ( −4 )$] ⓓ #math.equation(block: false, alt: "3 times 13 .")[$3 · 13 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−56")[$−56$] ⓑ 54 ⓒ #math.equation(block: false, alt: "−28")[$−28$] ⓓ 39
]
]

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by #math.equation(block: false, alt: "−1 ?")[$−1 ?$] Let’s multiply a positive number and then a negative number by #math.equation(block: false, alt: "−1")[$−1$] to see what we get.

#math.equation(block: true, alt: "−1 times 4, −1 open parenthesis −3 close parenthesis; Multiply., −4, 3; −4 is the opposite of 4 ., 3 is the opposite of −3 .")[$&  &  & −1 · 4 &  &  & −1 ( −3 ) \ "Multiply." &  &  & −4 &  &  & 3 \  &  &  & −4 #h(0.2em) "is the opposite of" #h(0.2em) 4 . &  &  & 3 #h(0.2em) "is the opposite of" #h(0.2em) −3 .$]Each time we multiply a number by #math.equation(block: false, alt: "−1 ,")[$−1 ,$] we get its opposite!

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication by #math.equation(block: false, alt: "−1")[$−1$]]

#math.equation(block: true, alt: "−1 a equals − a")[$−1 a = "−" a$]Multiplying a number by #math.equation(block: false, alt: "−1")[$−1$] gives its opposite.
]

#examplebox("Example 2")[][
Multiply: ⓐ #math.equation(block: false, alt: "−1 times 7")[$−1 · 7$] ⓑ #math.equation(block: false, alt: "−1 open parenthesis −11 close parenthesis .")[$−1 ( −11 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ  #linebreak() Multiply, noting that the signs are different so the product is negative.], [#math.equation(block: false, alt: "−1 times 7; −7; −7 is the opposite of 7 .")[$−1 · 7 \ −7 \ −7 #h(0.2em) "is the opposite of" #h(0.2em) 7 .$]]),
  [ⓑ  #linebreak() Multiply, noting that the signs are the same so the product is positive.], [#math.equation(block: false, alt: "−1 open parenthesis −11 close parenthesis; 11; 11 is the opposite of −11 .")[$−1 ( −11 ) \ 11 \ 11 #h(0.2em) "is the opposite of" #h(0.2em) −11 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: ⓐ #math.equation(block: false, alt: "−1 times 9")[$−1 · 9$] ⓑ #math.equation(block: false, alt: "−1 times open parenthesis −17 close parenthesis .")[$−1 · ( −17 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−9")[$−9$] ⓑ 17
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: ⓐ #math.equation(block: false, alt: "−1 times 8")[$−1 · 8$] ⓑ #math.equation(block: false, alt: "−1 times open parenthesis −16 close parenthesis .")[$−1 · ( −16 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−8")[$−8$] ⓑ 16
]
]


=== Divide Integers

What about #strong[division]? Division is the inverse operation of multiplication. So, #math.equation(block: false, alt: "15 divided by 3 equals 5")[$15 ÷ 3 = 5$] because #math.equation(block: false, alt: "5 times 3 equals 15 .")[$5 · 3 = 15 .$] In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

#math.equation(block: true, alt: "5 times 3, equals, 15 so 15 divided by 3, equals, 5, −5 open parenthesis 3 close parenthesis, equals, −15 so −15 divided by 3, equals, −5; open parenthesis −5 close parenthesis open parenthesis −3 close parenthesis, equals, 15 so 15 divided by open parenthesis −3 close parenthesis, equals, −5, 5 open parenthesis −3 close parenthesis, equals, −15 so −15 divided by open parenthesis −3 close parenthesis, equals, 5")[$5 · 3 & = & 15 #h(0.2em) "so" #h(0.2em) 15 ÷ 3 & = & 5 &  &  &  &  & −5 ( 3 ) & = & −15 #h(0.2em) "so" #h(0.2em) −15 ÷ 3 & = & −5 \ ( −5 ) ( −3 ) & = & 15 #h(0.2em) "so" #h(0.2em) 15 ÷ ( −3 ) & = & −5 &  &  &  &  & 5 ( −3 ) & = & −15 #h(0.2em) "so" #h(0.2em) −15 ÷ ( −3 ) & = & 5$]Division follows the same rules as multiplication!

For division of two signed numbers, when the:

- signs are the #emph[same], the quotient is #emph[positive].
- signs are #emph[different], the quotient is #emph[negative].

And remember that we can always check the answer of a division problem by multiplying.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication and Division of Signed Numbers]

For multiplication and division of two signed numbers:

- If the signs are the same, the result is positive.
- If the signs are different, the result is negative.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [Same signs], [Result],
  [Two positives #linebreak() Two negatives], [Positive #linebreak() Positive],
  [If the signs are the same, the result is positive.], [],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [Different signs], [Result],
  [Positive and negative #linebreak() Negative and positive], [Negative #linebreak() Negative],
  [If the signs are different, the result is negative.], [],
))
]

#examplebox("Example 3")[][
Divide: ⓐ #math.equation(block: false, alt: "−27 divided by 3")[$−27 ÷ 3$] ⓑ #math.equation(block: false, alt: "−100 divided by open parenthesis −4 close parenthesis .")[$−100 ÷ ( −4 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ  #linebreak() Divide. With different signs, the quotient is negative.], [#math.equation(block: false, alt: "minus 27 divided by 3; −9")[$− 27 ÷ 3 \ −9$]]),
  [ⓑ  #linebreak() Divide. With signs that are the same, the quotient is positive.], [#math.equation(block: false, alt: "minus 100 divided by open parenthesis −4 close parenthesis; 25")[$− 100 ÷ ( −4 ) \ 25$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: ⓐ #math.equation(block: false, alt: "−42 divided by 6")[$−42 ÷ 6$] ⓑ #math.equation(block: false, alt: "−117 divided by open parenthesis −3 close parenthesis .")[$−117 ÷ ( −3 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−7")[$−7$] ⓑ 39
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide: ⓐ #math.equation(block: false, alt: "−63 divided by 7")[$−63 ÷ 7$] ⓑ #math.equation(block: false, alt: "−115 divided by open parenthesis −5 close parenthesis .")[$−115 ÷ ( −5 ) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−9")[$−9$] ⓑ 23
]
]


=== Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "7 open parenthesis −2 close parenthesis plus 4 open parenthesis −7 close parenthesis minus 6 .")[$7 ( −2 ) + 4 ( −7 ) − 6 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "7 open parenthesis −2 close parenthesis plus 4 open parenthesis −7 close parenthesis minus 6")[$7 ( −2 ) + 4 ( −7 ) − 6$]]),
  [Multiply first.], [#math.equation(block: false, alt: "−14 plus open parenthesis −28 close parenthesis minus 6")[$−14 + ( −28 ) − 6$]],
  [Add.], [#math.equation(block: false, alt: "−42 minus 6")[$−42 − 6$]],
  [Subtract.], [#math.equation(block: false, alt: "−48")[$−48$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "8 open parenthesis −3 close parenthesis plus 5 open parenthesis −7 close parenthesis minus 4 .")[$8 ( −3 ) + 5 ( −7 ) − 4 .$]

#solutionbox[
#math.equation(block: true, alt: "−63")[$−63$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "9 open parenthesis −3 close parenthesis plus 7 open parenthesis −8 close parenthesis minus 1 .")[$9 ( −3 ) + 7 ( −8 ) − 1 .$]

#solutionbox[
#math.equation(block: true, alt: "−84")[$−84$]
]
]

#examplebox("Example 5")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power 4")[$attach(( −2 ), t: 4)$] ⓑ #math.equation(block: false, alt: "− 2 to the power 4 .")[$"−" 2^(4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
     #linebreak() Write in expanded form.
     #linebreak() Multiply.
     #linebreak() Multiply.
     #linebreak() Multiply.], [#math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power 4; open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis; 4 open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis; −8 open parenthesis −2 close parenthesis; 16")[$attach(( −2 ), t: 4) \ ( −2 ) ( −2 ) ( −2 ) ( −2 ) \ 4 ( −2 ) ( −2 ) \ −8 ( −2 ) \ 16$]]),
  [ⓑ 
     #linebreak() Write in expanded form. We are asked to find the opposite of#math.equation(block: false, alt: "2 to the power 4 .")[$#h(0.2em) 2^(4) .$]
	 #linebreak() Multiply.
	 #linebreak() Multiply.
	 #linebreak() Multiply.], [#math.equation(block: false, alt: "− 2 to the power 4; − open parenthesis 2 times 2 times 2 times 2 close parenthesis; − open parenthesis 4 times 2 times 2 close parenthesis; − open parenthesis 8 times 2 close parenthesis; -16")[$"−" 2^(4) \ "−" ( 2 · 2 · 2 · 2 ) \ "−" ( 4 · 2 · 2 ) \ "−" ( 8 · 2 ) \ -16$]],
))

Notice the difference in parts ⓐ and ⓑ. In part ⓐ , the exponent means to raise what is in the parentheses, the #math.equation(block: false, alt: "open parenthesis −2 close parenthesis")[$( −2 )$] to the #math.equation(block: false, alt: "4 to the power th")[$4^("th")$] power. In part ⓑ , the exponent means to raise just the 2 to the #math.equation(block: false, alt: "4 to the power th")[$4^("th")$] power and then take the opposite.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 4")[$attach(( −3 ), t: 4)$] ⓑ #math.equation(block: false, alt: "− 3 to the power 4 .")[$"−" 3^(4) .$]

#solutionbox[
ⓐ 81 ⓑ #math.equation(block: false, alt: "−81")[$−81$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −7 close parenthesis squared")[$attach(( −7 ), t: 2)$] ⓑ #math.equation(block: false, alt: "− 7 squared .")[$"−" 7^(2) .$]

#solutionbox[
ⓐ 49 ⓑ #math.equation(block: false, alt: "−49")[$−49$]
]
]

The next example reminds us to simplify inside parentheses first.

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "12 minus 3 open parenthesis 9 minus 12 close parenthesis .")[$12 − 3 ( 9 − 12 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "12 minus 3 open parenthesis 9 minus 12 close parenthesis")[$12 − 3 ( 9 − 12 )$]]),
  [Subtract in parentheses first.], [#math.equation(block: false, alt: "12 minus 3 open parenthesis −3 close parenthesis")[$12 − 3 ( −3 )$]],
  [Multiply.], [#math.equation(block: false, alt: "12 minus open parenthesis −9 close parenthesis")[$12 − ( −9 )$]],
  [Subtract.], [#math.equation(block: false, alt: "21")[$21$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "17 minus 4 open parenthesis 8 minus 11 close parenthesis .")[$17 − 4 ( 8 − 11 ) .$]

#solutionbox[
29
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "16 minus 6 open parenthesis 7 minus 13 close parenthesis .")[$16 − 6 ( 7 − 13 ) .$]

#solutionbox[
52
]
]

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −2 close parenthesis cubed .")[$8 ( −9 ) ÷ attach(( −2 ), t: 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −2 close parenthesis cubed")[$8 ( −9 ) ÷ attach(( −2 ), t: 3)$]]),
  [Exponents first.], [#math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −8 close parenthesis")[$8 ( −9 ) ÷ ( −8 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−72 divided by open parenthesis −8 close parenthesis")[$−72 ÷ ( −8 )$]],
  [Divide.], [#math.equation(block: false, alt: "9")[$9$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "12 open parenthesis −9 close parenthesis divided by open parenthesis −3 close parenthesis cubed .")[$12 ( −9 ) ÷ attach(( −3 ), t: 3) .$]

#solutionbox[
4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "18 open parenthesis −4 close parenthesis divided by open parenthesis −2 close parenthesis cubed .")[$18 ( −4 ) ÷ attach(( −2 ), t: 3) .$]

#solutionbox[
9
]
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "−30 divided by 2 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis .")[$−30 ÷ 2 + ( −3 ) ( −7 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−30 divided by 2 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis")[$−30 ÷ 2 + ( −3 ) ( −7 )$]]),
  [Multiply and divide left to right, so divide first.], [#math.equation(block: false, alt: "−15 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis")[$−15 + ( −3 ) ( −7 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−15 plus 21")[$−15 + 21$]],
  [Add.], [#math.equation(block: false, alt: "6")[$6$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−27 divided by 3 plus open parenthesis −5 close parenthesis open parenthesis −6 close parenthesis .")[$−27 ÷ 3 + ( −5 ) ( −6 ) .$]

#solutionbox[
21
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−32 divided by 4 plus open parenthesis −2 close parenthesis open parenthesis −7 close parenthesis .")[$−32 ÷ 4 + ( −2 ) ( −7 ) .$]

#solutionbox[
6
]
]


=== Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

#examplebox("Example 9")[][
When #math.equation(block: false, alt: "n equals −5 ,")[$n = −5 ,$] evaluate: ⓐ #math.equation(block: false, alt: "n plus 1")[$n + 1$] ⓑ #math.equation(block: false, alt: "− n plus 1 .")[$"−" n + 1 .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression 'n+1' is displayed, showing the variable 'n' incremented by one. This notation is commonly used in mathematics and computer science to represent the next integer or the successor of 'n'.], alt: "The mathematical expression 'n+1' is displayed, showing the variable 'n' incremented by one. This notation is commonly used in mathematics and computer science to represent the next integer or the successor of 'n'.", caption: none)]),
  [#figure(figph[Substitute -5 for n.], alt: "Substitute -5 for n.", caption: none)], [#figure(figph[A mathematical expression showing '-5 + 1' with the negative sign and the number 5 in red, and the plus sign and the number 1 in black.], alt: "A mathematical expression showing '-5 + 1' with the negative sign and the number 5 in red, and the plus sign and the number 1 in black.", caption: none)],
  [Simplify.], [−4],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression reads '-n+1' displayed on a white background.], alt: "A mathematical expression reads '-n+1' displayed on a white background.", caption: none)]),
  [#figure(figph[The image displays mathematical instructions, reading 'Substitute -5 for n.' The text is rendered in a black sans-serif font against a white background, with the number '-5' highlighted in red.], alt: "The image displays mathematical instructions, reading 'Substitute -5 for n.' The text is rendered in a black sans-serif font against a white background, with the number '-5' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression showing -(-5)+1.], alt: "A mathematical expression showing -(-5)+1.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression '5+1' is displayed in black text on a plain white background.], alt: "The mathematical expression '5+1' is displayed in black text on a plain white background.", caption: none)],
  [Add.], [6],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
When #math.equation(block: false, alt: "n equals −8 ,")[$n = −8 ,$] evaluate ⓐ #math.equation(block: false, alt: "n plus 2")[$n + 2$] ⓑ #math.equation(block: false, alt: "− n plus 2 .")[$"−" n + 2 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−6")[$−6$] ⓑ 10
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
When #math.equation(block: false, alt: "y equals −9 ,")[$y = −9 ,$] evaluate ⓐ #math.equation(block: false, alt: "y plus 8")[$y + 8$] ⓑ #math.equation(block: false, alt: "− y plus 8 .")[$"−" y + 8 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−1")[$−1$] ⓑ 17
]
]

#examplebox("Example 10")[][
Evaluate #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when #math.equation(block: false, alt: "x equals −18")[$x = −18$] and #math.equation(block: false, alt: "y equals 24 .")[$y = 24 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays a binomial (x + y) enclosed in parentheses, raised to the power of 2, signifying (x+y) squared.], alt: "A mathematical expression displays a binomial (x + y) enclosed in parentheses, raised to the power of 2, signifying (x+y) squared.", caption: none)]),
  [#figure(figph[The image shows the text 'Substitute -18 for x and 24 for y.', with '-18' in red and '24' in light blue, suggesting specific values for variables in an algebraic context.], alt: "The image shows the text 'Substitute -18 for x and 24 for y.', with '-18' in red and '24' in light blue, suggesting specific values for variables in an algebraic context.", caption: none)], [#figure(figph[The image shows the mathematical expression '(-18 + 24)^2', with the number -18 in red, 24 in light blue, and the rest of the equation in black.], alt: "The image shows the mathematical expression '(-18 + 24)^2', with the number -18 in red, 24 in light blue, and the rest of the equation in black.", caption: none)],
  [Add inside parenthesis.], [(6)#super[2]],
  [Simplify.], [36],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when #math.equation(block: false, alt: "x equals −15")[$x = −15$] and #math.equation(block: false, alt: "y equals 29 .")[$y = 29 .$]

#solutionbox[
196
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis cubed")[$attach(( x + y ), t: 3)$] when #math.equation(block: false, alt: "x equals −8")[$x = −8$] and #math.equation(block: false, alt: "y equals 10 .")[$y = 10 .$]

#solutionbox[
8
]
]

#examplebox("Example 11")[][
Evaluate #math.equation(block: false, alt: "20 minus z")[$20 − z$] when ⓐ #math.equation(block: false, alt: "z equals 12")[$z = 12$] and ⓑ #math.equation(block: false, alt: "z equals −12 .")[$z = −12 .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays a mathematical expression '20-z' in a bold, sans-serif font against a plain white background, representing twenty minus z.], alt: "The image displays a mathematical expression '20-z' in a bold, sans-serif font against a plain white background, representing twenty minus z.", caption: none)]),
  [#figure(figph[The text], alt: "The text", caption: none)], [#figure(figph[The image displays the numbers '20 - 12' with '20' in grey and '- 12' in red.], alt: "The image displays the numbers '20 - 12' with '20' in grey and '- 12' in red.", caption: none)],
  [Subtract.], [8],
))

#linebreak() 
ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the text '2Q-Z' in a dark gray font against a plain white background.], alt: "The image displays the text '2Q-Z' in a dark gray font against a plain white background.", caption: none)]),
  [#figure(figph[The text 'Substitute -12 for z.' is displayed, with the number '-12' highlighted in a reddish-orange color, suggesting an instruction for a mathematical or variable substitution task.], alt: "The text 'Substitute -12 for z.' is displayed, with the number '-12' highlighted in a reddish-orange color, suggesting an instruction for a mathematical or variable substitution task.", caption: none)], [#figure(figph[A mathematical expression '20 - (-12)' is shown, with the number -12 highlighted in red, indicating a subtraction of a negative number. This operation is equivalent to 20 + 12.], alt: "A mathematical expression '20 - (-12)' is shown, with the number -12 highlighted in red, indicating a subtraction of a negative number. This operation is equivalent to 20 + 12.", caption: none)],
  [Subtract.], [32],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "17 minus k")[$17 − k$] when ⓐ #math.equation(block: false, alt: "k equals 19")[$k = 19$] and ⓑ #math.equation(block: false, alt: "k equals −19 .")[$k = −19 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ 36
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "−5 minus b")[$−5 − b$] when ⓐ #math.equation(block: false, alt: "b equals 14")[$b = 14$] and ⓑ #math.equation(block: false, alt: "b equals −14 .")[$b = −14 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−19")[$−19$] ⓑ 9
]
]

#examplebox("Example 12")[][
Evaluate: #math.equation(block: false, alt: "2 x squared plus 3 x plus 8")[$2 x^(2) + 3 x + 8$] when #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$]

#solutionbox[
Substitute #math.equation(block: false, alt: "4 for x .")[$4 #h(0.2em) "for" #h(0.2em) x .$] Use parentheses to show multiplication. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression 2x^2 + 3x + 8.], alt: "The mathematical expression 2x^2 + 3x + 8.", caption: none)]),
  [Substitute.], [#figure(figph[A mathematical expression reads 2(4) squared + 3(4) + 8, with the number 4 highlighted in red each time it appears within the parentheses.], alt: "A mathematical expression reads 2(4) squared + 3(4) + 8, with the number 4 highlighted in red each time it appears within the parentheses.", caption: none)],
  [Evaluate exponents.], [#figure(figph[A mathematical expression showing the sum of products: 2 multiplied by 16, plus 3 multiplied by 4, plus 8. The expression is written as 2(16) + 3(4) + 8.], alt: "A mathematical expression showing the sum of products: 2 multiplied by 16, plus 3 multiplied by 4, plus 8. The expression is written as 2(16) + 3(4) + 8.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression showing the sum of three numbers: 32, 12, and 8, written as 32 + 12 + 8.], alt: "A mathematical expression showing the sum of three numbers: 32, 12, and 8, written as 32 + 12 + 8.", caption: none)],
  [Add.], [52],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "3 x squared minus 2 x plus 6")[$3 x^(2) − 2 x + 6$] when #math.equation(block: false, alt: "x equals −3 .")[$x = −3 .$]

#solutionbox[
39
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "4 x squared minus x minus 5")[$4 x^(2) − x − 5$] when #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$]

#solutionbox[
13
]
]


=== Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

#examplebox("Example 13")[][
Translate and simplify: the sum of 8 and #math.equation(block: false, alt: "−12 ,")[$−12 ,$] increased by 3.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [the #emph[sum] of 8 and #math.equation(block: false, alt: "−12 ,")[$−12 ,$] increased by 3.]),
  [Translate.], [#math.equation(block: false, alt: "[ 8 plus open parenthesis −12 close parenthesis ] plus 3")[$[ 8 + ( −12 ) ] + 3$]],
  [Simplify. Be careful not to confuse the brackets with an absolute value sign.], [#math.equation(block: false, alt: "open parenthesis −4 close parenthesis plus 3")[$( −4 ) + 3$]],
  [Add.], [#math.equation(block: false, alt: "−1")[$−1$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and simplify the sum of 9 and #math.equation(block: false, alt: "−16 ,")[$−16 ,$] increased by 4.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 9 plus open parenthesis −16 close parenthesis close parenthesis plus 4 ; minus 3")[$( 9 + ( −16 ) ) + 4 ; − 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and simplify the sum of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "−12 ,")[$−12 ,$] increased by 7.

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −8 plus open parenthesis −12 close parenthesis close parenthesis plus 7 ; minus 13")[$( −8 + ( −12 ) ) + 7 ; − 13$]
]
]

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "a minus b")[$a − b$]],
  [#math.equation(block: false, alt: "a")[$a$] minus #math.equation(block: false, alt: "b")[$b$] #linebreak() 
the difference of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] subtracted from #math.equation(block: false, alt: "a")[$a$] #linebreak() #math.equation(block: false, alt: "b")[$b$] less than #math.equation(block: false, alt: "a")[$a$]],
))

Be careful to get #emph[a] and #emph[b] in the right order!

#examplebox("Example 14")[][
Translate and then simplify ⓐ the difference of 13 and #math.equation(block: false, alt: "−21")[$−21$] ⓑ subtract 24 from #math.equation(block: false, alt: "−19 .")[$−19 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
     #linebreak() Translate.
     #linebreak() Simplify.], [#math.equation(block: false, alt: "the difference of 13 and minus 21; 13 minus open parenthesis −21 close parenthesis; 34")[$"the" #h(0.2em) "difference" #h(0.2em) "of" #h(0.2em) 13 #h(0.2em) "and" #h(0.2em) − 21 \ 13 − ( −21 ) \ 34$]]),
  [ⓑ  #linebreak() 
    Translate. Remember, "subtract #math.equation(block: false, alt: "b")[$b$] from #math.equation(block: false, alt: "a")[$a$] means #math.equation(block: false, alt: "a minus b")[$a − b$].
	 #linebreak() Simplify.], [#math.equation(block: false, alt: "subtract 24 from minus 19; minus 19 minus 24; minus 43")[$"subtract" #h(0.2em) 24 #h(0.2em) "from" #h(0.2em) − 19 \ − 19 − 24 \ − 43$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and simplify ⓐ the difference of 14 and #math.equation(block: false, alt: "−23")[$−23$] ⓑ subtract 21 from #math.equation(block: false, alt: "−17 .")[$−17 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "14 minus open parenthesis −23 close parenthesis ; 37")[$14 − ( −23 ) ; 37$] ⓑ #math.equation(block: false, alt: "−17 minus 21 ; minus 38")[$−17 − 21 ; − 38$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and simplify ⓐ the difference of 11 and #math.equation(block: false, alt: "−19")[$−19$] ⓑ subtract 18 from #math.equation(block: false, alt: "−11 .")[$−11 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "11 minus open parenthesis −19 close parenthesis ; 30")[$11 − ( −19 ) ; 30$] ⓑ #math.equation(block: false, alt: "−11 minus 18 ; minus 29")[$−11 − 18 ; − 29$]
]
]

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for #strong[multiplication] is “#strong[product]” and for #strong[division] is “#strong[quotient].”

#examplebox("Example 15")[][
Translate to an algebraic expression and simplify if possible: the product of #math.equation(block: false, alt: "−2")[$−2$] and 14.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the product of −2 and 14")[$"the product" #h(0.2em) "of" #h(0.2em) −2 #h(0.2em) "and" #h(0.2em) 14$]]),
  [Translate.], [#math.equation(block: false, alt: "open parenthesis −2 close parenthesis open parenthesis 14 close parenthesis")[$( −2 ) ( 14 )$]],
  [Simplify.], [#math.equation(block: false, alt: "−28")[$−28$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible: the product of #math.equation(block: false, alt: "−5")[$−5$] and 12.

#solutionbox[
#math.equation(block: true, alt: "−5 open parenthesis 12 close parenthesis ; minus 60")[$−5 ( 12 ) ; − 60$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible: the product of 8 and #math.equation(block: false, alt: "−13 .")[$−13 .$]

#solutionbox[
#math.equation(block: true, alt: "8 open parenthesis −13 close parenthesis ; minus 104")[$8 ( −13 ) ; − 104$]
]
]

#examplebox("Example 16")[][
Translate to an algebraic expression and simplify if possible: the quotient of #math.equation(block: false, alt: "−56")[$−56$] and #math.equation(block: false, alt: "−7 .")[$−7 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the quotient of −56 and −7")[$"the quotient" #h(0.2em) "of" #h(0.2em) −56 #h(0.2em) "and" #h(0.2em) −7$]]),
  [Translate.], [#math.equation(block: false, alt: "−56 divided by open parenthesis −7 close parenthesis")[$−56 ÷ ( −7 )$]],
  [Simplify.], [#math.equation(block: false, alt: "8")[$8$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible: the quotient of #math.equation(block: false, alt: "−63")[$−63$] and #math.equation(block: false, alt: "−9 .")[$−9 .$]

#solutionbox[
#math.equation(block: true, alt: "−63 divided by open parenthesis −9 close parenthesis ; 7")[$−63 ÷ ( −9 ) ; 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible: the quotient of #math.equation(block: false, alt: "−72")[$−72$] and #math.equation(block: false, alt: "−9 .")[$−9 .$]

#solutionbox[
#math.equation(block: true, alt: "−72 divided by open parenthesis −9 close parenthesis ; 8")[$−72 ÷ ( −9 ) ; 8$]
]
]


=== Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

#examplebox("Example 17")[How to Apply a Strategy to Solve Applications with Integers][
In the morning, the temperature in Urbana, Illinois was 11 degrees. By mid-afternoon, the temperature had dropped to #math.equation(block: false, alt: "−9")[$−9$] degrees. What was the difference of the morning and afternoon temperatures?

#solutionbox[
#figure(figph[This is a table with two columns. The left column includes steps to solve the problem. The right column includes the math to solve the problem. In the first row, the left column says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The right column is blank.], alt: "This is a table with two columns. The left column includes steps to solve the problem. The right column includes the math to solve the problem. In the first row, the left column says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The right column is blank.", caption: none)

#figure(figph[In the second row, the left column says “Step 2. Identify what we are asked to find”. The right column says, “the difference of the morning  and afternoon temperatures.”], alt: "In the second row, the left column says “Step 2. Identify what we are asked to find”. The right column says, “the difference of the morning and afternoon temperatures.”", caption: none)

#figure(figph[In the third row, the left column says, “Step 3. Write a phrase that gives the information to find it.” Next to this in the right column, it says “the difference of 11 and negative 9.”], alt: "In the third row, the left column says, “Step 3. Write a phrase that gives the information to find it.” Next to this in the right column, it says “the difference of 11 and negative 9.”", caption: none)

#figure(figph[In the fourth row, the left column says, “Step 4. Translate the phrase to an expression.” The right column contains 11 minus negative 9.], alt: "In the fourth row, the left column says, “Step 4. Translate the phrase to an expression.” The right column contains 11 minus negative 9.", caption: none)

#figure(figph[In the fifth row, the left column says, “Step 5. Simplify the expression.” The right column contains 20.], alt: "In the fifth row, the left column says, “Step 5. Simplify the expression.” The right column contains 20.", caption: none)

#figure(figph[The final row says, “Step five. Write a complete sentence that answers the question.” Next to this in the right column, it says “the difference in temperatures was 20 degrees.”], alt: "The final row says, “Step five. Write a complete sentence that answers the question.” Next to this in the right column, it says “the difference in temperatures was 20 degrees.”", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In the morning, the temperature in Anchorage, Alaska was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

#solutionbox[
The difference in temperatures was 45 degrees.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The temperature in Denver was #math.equation(block: false, alt: "−6")[$−6$] degrees at lunchtime. By sunset the temperature had dropped to #math.equation(block: false, alt: "−15")[$−15$] degrees. What was the difference in the lunchtime and sunset temperatures?

#solutionbox[
The difference in temperatures was 9 degrees.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Apply a Strategy to Solve Applications with Integers.]

+ Read the problem. Make sure all the words and ideas are understood
+ Identify what we are asked to find.
+ Write a phrase that gives the information to find it.
+ Translate the phrase to an expression.
+ Simplify the expression.
+ Answer the question with a complete sentence.
]

#examplebox("Example 18")[][
The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1.] Read the problem. Make sure all the words and ideas are understood.], []),
  [#emph[Step 2.] Identify what we are asked to find.], [the number of yards lost],
  [#emph[Step 3.] Write a phrase that gives the information to find it.], [three times a 15-yard penalty],
  [#emph[Step 4.] Translate the phrase to an expression.], [#math.equation(block: false, alt: "3 open parenthesis −15 close parenthesis")[$3 ( −15 )$]],
  [#emph[Step 5.] Simplify the expression.], [#math.equation(block: false, alt: "−45")[$−45$]],
  [#emph[Step 6.] Answer the question with a complete sentence.], [The team lost 45 yards.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

#solutionbox[
The Bears lost 105 yards.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a \$2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

#solutionbox[
A \$16 fee was deducted from his checking account.
]
]


=== Key Concepts

- #strong[Multiplication and Division of Two Signed Numbers]
- Same signs—Product is positive
- Different signs—Product is negative
- #strong[Strategy for Applications]
+ Identify what you are asked to find.
+ Write a phrase that gives the information to find it.
+ Translate the phrase to an expression.
+ Simplify the expression.
+ Answer the question with a complete sentence.


==== Practice Makes Perfect

#strong[Multiply Integers]

In the following exercises, multiply.

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

#solutionbox[
#math.equation(block: true, alt: "−32")[$−32$]
]

#math.equation(block: true, alt: "−3 times 9")[$−3 · 9$]

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

#solutionbox[
#math.equation(block: true, alt: "−63")[$−63$]
]

#math.equation(block: true, alt: "13 open parenthesis −5 close parenthesis")[$13 ( −5 )$]

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

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

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

#math.equation(block: true, alt: "−1 open parenthesis −14 close parenthesis")[$−1 ( −14 )$]

#solutionbox[
14
]

#math.equation(block: true, alt: "−1 open parenthesis −19 close parenthesis")[$−1 ( −19 )$]

#strong[Divide Integers]

In the following exercises, divide.

#math.equation(block: true, alt: "−24 divided by 6")[$−24 ÷ 6$]

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

#math.equation(block: true, alt: "35 divided by open parenthesis −7 close parenthesis")[$35 ÷ ( −7 )$]

#math.equation(block: true, alt: "−52 divided by open parenthesis −4 close parenthesis")[$−52 ÷ ( −4 )$]

#solutionbox[
13
]

#math.equation(block: true, alt: "−84 divided by open parenthesis −6 close parenthesis")[$−84 ÷ ( −6 )$]

#math.equation(block: true, alt: "−180 divided by 15")[$−180 ÷ 15$]

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

#math.equation(block: true, alt: "−192 divided by 12")[$−192 ÷ 12$]

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "5 open parenthesis −6 close parenthesis plus 7 open parenthesis −2 close parenthesis minus 3")[$5 ( −6 ) + 7 ( −2 ) − 3$]

#solutionbox[
#math.equation(block: true, alt: "−47")[$−47$]
]

#math.equation(block: true, alt: "8 open parenthesis −4 close parenthesis plus 5 open parenthesis −4 close parenthesis minus 6")[$8 ( −4 ) + 5 ( −4 ) − 6$]

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis to the power 6")[$attach(( −2 ), t: 6)$]

#solutionbox[
64
]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis to the power 5")[$attach(( −3 ), t: 5)$]

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

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

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

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

#solutionbox[
90
]

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

#math.equation(block: true, alt: "open parenthesis 8 minus 11 close parenthesis open parenthesis 9 minus 12 close parenthesis")[$( 8 − 11 ) ( 9 − 12 )$]

#solutionbox[
9
]

#math.equation(block: true, alt: "open parenthesis 6 minus 11 close parenthesis open parenthesis 8 minus 13 close parenthesis")[$( 6 − 11 ) ( 8 − 13 )$]

#math.equation(block: true, alt: "26 minus 3 open parenthesis 2 minus 7 close parenthesis")[$26 − 3 ( 2 − 7 )$]

#solutionbox[
41
]

#math.equation(block: true, alt: "23 minus 2 open parenthesis 4 minus 6 close parenthesis")[$23 − 2 ( 4 − 6 )$]

#math.equation(block: true, alt: "65 divided by open parenthesis −5 close parenthesis plus open parenthesis −28 close parenthesis divided by open parenthesis −7 close parenthesis")[$65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )$]

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

#math.equation(block: true, alt: "52 divided by open parenthesis −4 close parenthesis plus open parenthesis −32 close parenthesis divided by open parenthesis −8 close parenthesis")[$52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )$]

#math.equation(block: true, alt: "9 minus 2 [ 3 minus 8 open parenthesis −2 close parenthesis ]")[$9 − 2 [ 3 − 8 ( −2 ) ]$]

#solutionbox[
#math.equation(block: true, alt: "−29")[$−29$]
]

#math.equation(block: true, alt: "11 minus 3 [ 7 minus 4 open parenthesis −2 close parenthesis ]")[$11 − 3 [ 7 − 4 ( −2 ) ]$]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis squared minus 24 divided by open parenthesis 8 minus 2 close parenthesis")[$attach(( −3 ), t: 2) − 24 ÷ ( 8 − 2 )$]

#solutionbox[
5
]

#math.equation(block: true, alt: "open parenthesis −4 close parenthesis squared minus 32 divided by open parenthesis 12 minus 4 close parenthesis")[$attach(( −4 ), t: 2) − 32 ÷ ( 12 − 4 )$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: false, alt: "y plus open parenthesis −14 close parenthesis")[$y + ( −14 )$] when ⓐ #math.equation(block: false, alt: "y equals −33")[$y = −33$] ⓑ #math.equation(block: false, alt: "y equals 30")[$y = 30$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−47")[$−47$] ⓑ 16
]

#math.equation(block: false, alt: "x plus open parenthesis −21 close parenthesis")[$x + ( −21 )$] whenⓐ #math.equation(block: false, alt: "x equals −27")[$x = −27$]ⓑ #math.equation(block: false, alt: "x equals 44")[$x = 44$]

+ ⓐ #math.equation(block: false, alt: "a plus 3")[$a + 3$] when #math.equation(block: false, alt: "a equals −7")[$a = −7$]
+ ⓑ #math.equation(block: false, alt: "− a plus 3")[$"−" a + 3$] when #math.equation(block: false, alt: "a equals −7")[$a = −7$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ 10
]

+ ⓐ #math.equation(block: false, alt: "d plus open parenthesis −9 close parenthesis")[$d + ( −9 )$] when #math.equation(block: false, alt: "d equals −8")[$d = −8$]
+ ⓑ #math.equation(block: false, alt: "− d plus open parenthesis −9 close parenthesis")[$"−" d + ( −9 )$] when #math.equation(block: false, alt: "d equals −8")[$d = −8$]

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

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

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

#math.equation(block: false, alt: "r plus s")[$r + s$] when #math.equation(block: false, alt: "r equals −9 , s equals −7")[$r = −9 , s = −7$]

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

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

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

#solutionbox[
121
]

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

#math.equation(block: false, alt: "−2 x plus 17")[$−2 x + 17$] when

+ ⓐ #math.equation(block: false, alt: "x equals 8")[$x = 8$]
+ ⓑ #math.equation(block: false, alt: "x equals −8")[$x = −8$]

#solutionbox[
+ ⓐ 1
+ ⓑ 33
]

#math.equation(block: false, alt: "−5 y plus 14")[$−5 y + 14$] when #linebreak() ⓐ #math.equation(block: false, alt: "y equals 9")[$y = 9$] #linebreak() ⓑ #math.equation(block: false, alt: "y equals −9")[$y = −9$]

#math.equation(block: false, alt: "10 minus 3 m")[$10 − 3 m$] when #linebreak() ⓐ #math.equation(block: false, alt: "m equals 5")[$m = 5$] #linebreak() ⓑ #math.equation(block: false, alt: "m equals −5")[$m = −5$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−5")[$−5$] ⓑ 25
]

#math.equation(block: false, alt: "18 minus 4 n")[$18 − 4 n$] when #linebreak() ⓐ #math.equation(block: false, alt: "n equals 3")[$n = 3$] #linebreak() ⓑ #math.equation(block: false, alt: "n equals −3")[$n = −3$]

#math.equation(block: false, alt: "2 w squared minus 3 w plus 7")[$2 w^(2) − 3 w + 7$] when #linebreak() #math.equation(block: false, alt: "w equals −2")[$w = −2$]

#solutionbox[
21
]

#math.equation(block: false, alt: "3 u squared minus 4 u plus 5")[$3 u^(2) − 4 u + 5$] when #math.equation(block: false, alt: "u equals −3")[$u = −3$]

#math.equation(block: false, alt: "9 a minus 2 b minus 8")[$9 a − 2 b − 8$] when #linebreak() #math.equation(block: false, alt: "a equals −6 and b equals −3")[$a = −6 #h(0.2em) "and" #h(0.2em) b = −3$]

#solutionbox[
#math.equation(block: true, alt: "−56")[$−56$]
]

#math.equation(block: false, alt: "7 m minus 4 n minus 2")[$7 m − 4 n − 2$] when #linebreak() #math.equation(block: false, alt: "m equals −4 and n equals −9")[$m = −4 #h(0.2em) "and" #h(0.2em) n = −9$]

#strong[Translate English Phrases to Algebraic Expressions]

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

the sum of 3 and #math.equation(block: false, alt: "−15 ,")[$−15 ,$] increased by 7

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 plus open parenthesis −15 close parenthesis close parenthesis plus 7 ; minus 5")[$( 3 + ( −15 ) ) + 7 ; − 5$]
]

the sum of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "−9 ,")[$−9 ,$] increased by 23

the difference of 10 and #math.equation(block: false, alt: "−18")[$−18$]

#solutionbox[
#math.equation(block: true, alt: "10 minus open parenthesis −18 close parenthesis ; 28")[$10 − ( −18 ) ; 28$]
]

subtract 11 from #math.equation(block: false, alt: "−25")[$−25$]

the difference of #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "−30")[$−30$]

#solutionbox[
#math.equation(block: true, alt: "−5 minus open parenthesis −30 close parenthesis ; 25")[$−5 − ( −30 ) ; 25$]
]

subtract #math.equation(block: false, alt: "−6")[$−6$] from #math.equation(block: false, alt: "−13")[$−13$]

the product of #math.equation(block: false, alt: "−3 and 15")[$"−3 and 15"$]

#solutionbox[
#math.equation(block: true, alt: "−3 times 15 ; minus 45")[$−3 · 15 ; − 45$]
]

the product of #math.equation(block: false, alt: "−4 and 16")[$"−4 and 16" #h(0.2em)$]

the quotient of #math.equation(block: false, alt: "−60")[$−60$] and #math.equation(block: false, alt: "−20")[$−20$]

#solutionbox[
#math.equation(block: true, alt: "−60 divided by open parenthesis −20 close parenthesis ; 3")[$−60 ÷ ( −20 ) ; 3$]
]

the quotient of #math.equation(block: false, alt: "−40")[$−40$] and #math.equation(block: false, alt: "−20")[$−20$]

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

#solutionbox[
#math.equation(block: true, alt: "the fraction −6 over a plus b")[$frac(−6, a + b)$]
]

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

the product of #math.equation(block: false, alt: "−10")[$−10$] and the difference of #math.equation(block: false, alt: "p and q")[$p #h(0.2em) "and" #h(0.2em) q$]

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

the product of #math.equation(block: false, alt: "−13")[$−13$] and the difference of #math.equation(block: false, alt: "c and d")[$c #h(0.2em) "and" #h(0.2em) d$]

#strong[Use Integers in Applications]

In the following exercises, solve.

#strong[Temperature] On January #math.equation(block: false, alt: "15 ,")[$15 ,$] the high temperature in Anaheim, California, was #math.equation(block: false, alt: "84 ° .")[$84 "°" .$] That same day, the high temperature in Embarrass, Minnesota was #math.equation(block: false, alt: "−12 ° .")[$−12 "°" .$] What was the difference between the temperature in Anaheim and the temperature in Embarrass?

#solutionbox[
#math.equation(block: true, alt: "96 °")[$96 "°"$]
]

#strong[Temperature] On January #math.equation(block: false, alt: "21 ,")[$21 ,$] the high temperature in Palm Springs, California, was #math.equation(block: false, alt: "89 ° ,")[$89 "°" ,$] and the high temperature in Whitefield, New Hampshire was #math.equation(block: false, alt: "−31 ° .")[$−31 "°" .$] What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

#strong[Football] On the first down, the Chargers had the ball on their 25-yard line. They lost 6 yards on the first-down play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?

#solutionbox[
21
]

#strong[Football] On first down, the Steelers had the ball on their 30-yard line. They gained 9 yards on the first-down play, lost 14 yards on the second-down play, and lost 2 yards on the third-down play. What was the yard line at the end of the third-down play?

#strong[Checking Account] Mayra has \$124 in her checking account. She writes a check for \$152. What is the new balance in her checking account?

#solutionbox[
#math.equation(block: true, alt: "− $28")[$"−" \$28$]
]

#strong[Checking Account] Selina has \$165 in her checking account. She writes a check for \$207. What is the new balance in her checking account?

#strong[Checking Account] Diontre has a balance of #math.equation(block: false, alt: "− $38")[$"−" \$38$] in his checking account. He deposits \$225 to the account. What is the new balance?

#solutionbox[
\$187
]

#strong[Checking Account] Reymonte has a balance of #math.equation(block: false, alt: "− $49")[$"−" \$49$] in his checking account. He deposits \$281 to the account. What is the new balance?


==== Everyday Math

#strong[Stock market] Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped \$12 per share. What was the total effect on Javier’s portfolio?

#solutionbox[
#math.equation(block: true, alt: "− $3600")[$"−" \$3600$]
]

#strong[Weight loss] In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?


==== Writing Exercises

In your own words, state the rules for multiplying integers.

#solutionbox[
Answers may vary
]

In your own words, state the rules for dividing integers.

Why is #math.equation(block: false, alt: "− 2 to the power 4 not equal to open parenthesis −2 close parenthesis to the power 4 ?")[$"−" 2^(4) ≠ attach(( −2 ), t: 4) ?$]

#solutionbox[
Answers may vary
]

Why is #math.equation(block: false, alt: "− 4 cubed equals open parenthesis −4 close parenthesis cubed ?")[$"−" 4^(3) = attach(( −4 ), t: 3) ?$]


==== 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 that is composed of four columns and seven rows. The titles of the columns are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The first column reads “multiple integers.”, “divide integers.”, “simplify expressions with integers.”, “evaluate variable expressions with integers.”, “translate English phrases to algebraic expressions.” and “use integers in applications.”], alt: "A table is shown that is composed of four columns and seven rows. The titles of the columns are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The first column reads “multiple integers.”, “divide integers.”, “simplify expressions with integers.”, “evaluate variable expressions with integers.”, “translate English phrases to algebraic expressions.” and “use integers in applications.”", caption: none)

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