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== 7.4#h(0.6em)Properties of Identity, Inverses, and Zero

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

Find the opposite of #math.equation(block: false, alt: "−4 .")[$−4 .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the reciprocal of #math.equation(block: false, alt: "the fraction 5 over 2 .")[$frac(5, 2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 3 a over 5 times the fraction 9 over 2 a .")[$frac(3 a, 5) · frac(9, 2 a) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 27 over 10")[$frac(27, 10)$]
]
]


=== Recognize the Identity Properties of Addition and Multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call #math.equation(block: false, alt: "0")[$0$] the #strong[additive identity].

For example,

#math.equation(block: true, alt: "13 plus 0, , −14 plus 0, , 0 plus open parenthesis −3 x close parenthesis; 13, , −14, , −3 x")[$13 + 0 & #h(2em) & −14 + 0 & #h(2em) & 0 + ( −3 x ) \ 13 & #h(2em) & −14 & #h(2em) & −3 x$]What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call #math.equation(block: false, alt: "1")[$1$] the #strong[multiplicative identity].

For example,

#math.equation(block: true, alt: "43 times 1, , −27 times 1, , 1 times the fraction 6 y over 5; 43, , −27, , the fraction 6 y over 5")[$43 · 1 & #h(2em) & −27 · 1 & #h(2em) & 1 · frac(6 y, 5) \ 43 & #h(2em) & −27 & #h(2em) & frac(6 y, 5)$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Identity Properties]

The #emph[identity property of addition]: for any real number #math.equation(block: false, alt: "a ,")[$a ,$]

#math.equation(block: true, alt: "a plus 0 equals a 0 plus a equals a; 0 is called the additive identity")[$a + 0 = a #h(2em) 0 + a = a \ "0 is called the " #h(2em) " additive identity"$]The #emph[identity property of multiplication]: for any real number #math.equation(block: false, alt: "a")[$a$]

#math.equation(block: true, alt: "a times 1 equals a 1 times a equals a; 1 is called the multiplicative identity")[$a · 1 = a #h(2em) 1 · a = a \ "1 is called the " #h(2em) "multiplicative identity"$]
]

#examplebox("Example 1")[][
Identify whether each equation demonstrates the identity property of addition or multiplication.

+ ⓐ #math.equation(block: false, alt: "7 plus 0 equals 7")[$#h(0.2em) 7 + 0 = 7$]
+ ⓑ #math.equation(block: false, alt: "−16 open parenthesis 1 close parenthesis equals −16")[$#h(0.2em) −16 ( 1 ) = −16$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "7 plus 0 equals 7")[$7 + 0 = 7$]],
  [We are adding 0.], [We are using the identity property of addition.],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "−16 open parenthesis 1 close parenthesis equals −16")[$−16 ( 1 ) = −16$]],
  [We are multiplying by 1.], [We are using the identity property of multiplication.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify whether each equation demonstrates the identity property of addition or multiplication:

ⓐ #math.equation(block: false, alt: "23 plus 0 equals 23")[$#h(0.2em) 23 + 0 = 23$]ⓑ #math.equation(block: false, alt: "−37 open parenthesis 1 close parenthesis equals −37 .")[$#h(0.2em) −37 ( 1 ) = −37 .$]

#solutionbox[
+ ⓐ identity property of addition
+ ⓑ identity property of multiplication
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Identify whether each equation demonstrates the identity property of addition or multiplication:

ⓐ #math.equation(block: false, alt: "1 times 29 equals 29")[$#h(0.2em) 1 · 29 = 29$]ⓑ #math.equation(block: false, alt: "14 plus 0 equals 14 .")[$#h(0.2em) 14 + 0 = 14 .$]

#solutionbox[
+ ⓐ identity property of multiplication
+ ⓑ identity property of addition
]
]


=== Use the Inverse Properties of Addition and Multiplication

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [What number added to 5 gives the additive identity, 0?], [],
  [#math.equation(block: false, alt: "5 plus _____ equals 0")[$5 + \_\_\_\_\_ = 0$]], [#figure(figph[The text reads 'We know 5 + (-5) = 0' illustrating the concept of additive inverses where a number and its negative sum to zero.], alt: "The text reads 'We know 5 + (-5) = 0' illustrating the concept of additive inverses where a number and its negative sum to zero.", caption: none)],
  [What number added to −6 gives the additive identity, 0?], [],
  [#math.equation(block: false, alt: "−6 plus _____ equals 0")[$−6 + \_\_\_\_\_ = 0$]], [#figure(figph[The image displays the equation 'We know -6 + 6 = 0', written in a dark teal color, with the second '6' in the equation highlighted in red to emphasize its role in the sum.], alt: "The image displays the equation 'We know -6 + 6 = 0', written in a dark teal color, with the second '6' in the equation highlighted in red to emphasize its role in the sum.", caption: none)],
))

Notice that in each case, the missing number was the opposite of the number.

We call #math.equation(block: false, alt: "minus a")[$− a$] the #strong[additive inverse] of #math.equation(block: false, alt: "a .")[$a .$] The opposite of a number is its additive inverse. A number and its opposite add to #math.equation(block: false, alt: "0 ,")[$0 ,$] which is the additive identity.

What number multiplied by #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] gives the multiplicative identity, #math.equation(block: false, alt: "1 ?")[$1 ?$] In other words, two-thirds times what results in #math.equation(block: false, alt: "1 ?")[$1 ?$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "the fraction 2 over 3 times ___ equals 1")[$frac(2, 3) · \_\_\_ = 1$]], [#figure(figph[The image displays the equation 'We know (2/3) \* (3/2) = 1', demonstrating that a fraction multiplied by its reciprocal results in 1.], alt: "The image displays the equation 'We know (2/3) * (3/2) = 1', demonstrating that a fraction multiplied by its reciprocal results in 1.", caption: none)],
))

What number multiplied by #math.equation(block: false, alt: "2")[$2$] gives the multiplicative identity, #math.equation(block: false, alt: "1 ?")[$1 ?$] In other words two times what results in #math.equation(block: false, alt: "1 ?")[$1 ?$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#math.equation(block: false, alt: "2 times ___ equals 1")[$2 · \_\_\_ = 1$]], [#figure(figph[The equation illustrates that two multiplied by one half equals one.], alt: "The equation illustrates that two multiplied by one half equals one.", caption: none)],
))

Notice that in each case, the missing number was the reciprocal of the number.

We call #math.equation(block: false, alt: "the fraction 1 over a")[$frac(1, a)$] the #strong[multiplicative inverse] of #math.equation(block: false, alt: "a open parenthesis a not equal to 0 close parenthesis .")[$a ( a ≠ 0 ) "."$] The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to #math.equation(block: false, alt: "1 ,")[$1 ,$] which is the multiplicative identity.

We’ll formally state the Inverse Properties here:

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

#emph[Inverse Property of Addition] for any real number #math.equation(block: false, alt: "a ,")[$a ,$]

#math.equation(block: true, alt: "a plus open parenthesis minus a close parenthesis equals 0; minus a is the additive inverse of a .")[$a + ( − a ) = 0 \ − a #h(0.2em) "is the " #h(2em) "additive inverse" #h(2em) " of" #h(0.2em) a .$]#emph[Inverse Property of Multiplication] for any real number #math.equation(block: false, alt: "a not equal to 0 ,")[$a ≠ 0 ,$]

#math.equation(block: true, alt: "a times the fraction 1 over a equals 1; the fraction 1 over a is the multiplicative inverse of a .")[$a · frac(1, a) = 1 \ frac(1, a) #h(0.2em) "is the " #h(2em) "multiplicative inverse" #h(2em) " of" #h(0.2em) a .$]
]

#examplebox("Example 2")[][
Find the additive inverse of each expression: ⓐ #math.equation(block: false, alt: "13")[$13$] ⓑ #math.equation(block: false, alt: "minus the fraction 5 over 8")[$− frac(5, 8)$] ⓒ #math.equation(block: false, alt: "0.6")[$#h(0.2em) 0.6$].

#solutionbox[
To find the additive inverse, we find the opposite.

+ ⓐ The additive inverse of #math.equation(block: false, alt: "13")[$13$] is its opposite, #math.equation(block: false, alt: "−13 .")[$−13 .$]
+ ⓑ The additive inverse of #math.equation(block: false, alt: "minus the fraction 5 over 8")[$− frac(5, 8)$] is its opposite, #math.equation(block: false, alt: "the fraction 5 over 8 .")[$frac(5, 8) .$]
+ ⓒ The additive inverse of #math.equation(block: false, alt: "0.6")[$0.6$] is its opposite, #math.equation(block: false, alt: "−0.6 .")[$−0.6 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the additive inverse: ⓐ #math.equation(block: false, alt: "18")[$18$] ⓑ #math.equation(block: false, alt: "the fraction 7 over 9")[$frac(7, 9)$] ⓒ #math.equation(block: false, alt: "1.2")[$1.2$].

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "−18")[$#h(0.2em) −18 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 7 over 9")[$#h(0.2em) − frac(7, 9)$]
+ ⓒ #math.equation(block: false, alt: "−1.2")[$#h(0.2em) −1.2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the additive inverse: ⓐ #math.equation(block: false, alt: "47")[$47$] ⓑ #math.equation(block: false, alt: "the fraction 7 over 13")[$frac(7, 13)$] ⓒ #math.equation(block: false, alt: "8.4")[$#h(0.2em) 8.4$].

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "−47")[$#h(0.2em) −47$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 7 over 13")[$#h(0.2em) − frac(7, 13)$]
+ ⓒ #math.equation(block: false, alt: "−8.4")[$#h(0.2em) −8.4$]
]
]

#examplebox("Example 3")[][
Find the multiplicative inverse: ⓐ #math.equation(block: false, alt: "9")[$9 #h(0.2em)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 9")[$− frac(1, 9) #h(0.2em)$] ⓒ #math.equation(block: false, alt: "0.9")[$0.9$].

#solutionbox[
To find the multiplicative inverse, we find the reciprocal.

+ ⓐ The multiplicative inverse of #math.equation(block: false, alt: "9")[$9$] is its reciprocal, #math.equation(block: false, alt: "the fraction 1 over 9 .")[$frac(1, 9) .$]
+ ⓑ The multiplicative inverse of #math.equation(block: false, alt: "minus the fraction 1 over 9")[$− frac(1, 9)$] is its reciprocal, #math.equation(block: false, alt: "−9 .")[$−9 .$]
+ ⓒ To find the multiplicative inverse of #math.equation(block: false, alt: "0.9 ,")[$0.9 ,$] we first convert #math.equation(block: false, alt: "0.9")[$0.9$] to a fraction, #math.equation(block: false, alt: "the fraction 9 over 10 .")[$frac(9, 10) .$] Then we find the reciprocal, #math.equation(block: false, alt: "the fraction 10 over 9 .")[$frac(10, 9) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the multiplicative inverse: ⓐ #math.equation(block: false, alt: "5")[$#h(0.2em) 5 #h(0.2em)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 7")[$#h(0.2em) − frac(1, 7) #h(0.2em)$] ⓒ #math.equation(block: false, alt: "0.3")[$#h(0.2em) 0.3$].

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 5")[$#h(0.2em) frac(1, 5)$]
+ ⓑ #math.equation(block: false, alt: "−7")[$#h(0.2em) −7$]
+ ⓒ #math.equation(block: false, alt: "the fraction 10 over 3")[$#h(0.2em) frac(10, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the multiplicative inverse: ⓐ #math.equation(block: false, alt: "18")[$#h(0.2em) 18$] ⓑ #math.equation(block: false, alt: "minus the fraction 4 over 5")[$#h(0.2em) − frac(4, 5)$] ⓒ #math.equation(block: false, alt: "0.6")[$#h(0.2em) 0.6$].

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 1 over 18")[$#h(0.2em) frac(1, 18)$]
+ ⓑ #math.equation(block: false, alt: "minus the fraction 5 over 4")[$#h(0.2em) − frac(5, 4)$]
+ ⓒ #math.equation(block: false, alt: "the fraction 5 over 3")[$#h(0.2em) frac(5, 3)$]
]
]


=== Use the Properties of Zero

We have already learned that zero is the #strong[additive identity], since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.


==== Multiplication by Zero

What happens when you multiply a number by #math.equation(block: false, alt: "0 ?")[$0 ?$] Multiplying by #math.equation(block: false, alt: "0")[$0$] makes the product equal zero. The product of any real number and #math.equation(block: false, alt: "0")[$0$] is #math.equation(block: false, alt: "0 .")[$0 .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication by Zero]

For any real number #math.equation(block: false, alt: "a ,")[$a ,$]

#math.equation(block: true, alt: "a times 0 equals 0 0 times a equals 0")[$a · 0 = 0 #h(2em) 0 · a = 0$]
]

#examplebox("Example 4")[][
Simplify: ⓐ #math.equation(block: false, alt: "−8 times 0")[$#h(0.2em) −8 · 0$] ⓑ #math.equation(block: false, alt: "the fraction 5 over 12 times 0")[$#h(0.2em) frac(5, 12) · 0$] ⓒ #math.equation(block: false, alt: "0 open parenthesis 2.94 close parenthesis")[$#h(0.2em) 0 ( 2.94 )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "−8 times 0")[$−8 ⋅ 0$]],
  [The product of any real number and 0 is 0.], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "the fraction 5 over 12 times 0")[$frac(5, 12) · 0$]],
  [The product of any real number and 0 is 0.], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "0 open parenthesis 2.94 close parenthesis")[$0 ( 2.94 )$]],
  [The product of any real number and 0 is 0.], [#math.equation(block: false, alt: "0")[$0$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "−14 times 0")[$#h(0.2em) −14 · 0 #h(0.2em)$] ⓑ #math.equation(block: false, alt: "0 times the fraction 2 over 3")[$0 · frac(2, 3) #h(0.2em)$] ⓒ #math.equation(block: false, alt: "open parenthesis 16.5 close parenthesis times 0 .")[$#h(0.2em) ( 16.5 ) · 0 .$]

#solutionbox[
+ ⓐ 0
+ ⓑ 0
+ ⓒ 0
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 1.95 close parenthesis times 0")[$#h(0.2em) ( 1.95 ) · 0$] ⓑ #math.equation(block: false, alt: "0 open parenthesis −17 close parenthesis")[$#h(0.2em) 0 ( −17 )$] ⓒ #math.equation(block: false, alt: "0 times the fraction 5 over 4 .")[$#h(0.2em) 0 · frac(5, 4) .$]

#solutionbox[
+ ⓐ 0
+ ⓑ 0
+ ⓒ 0
]
]


==== Dividing with Zero

What about dividing with #math.equation(block: false, alt: "0 ?")[$0 ?$] Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are #math.equation(block: false, alt: "0")[$0$] cookies to share, so each person gets #math.equation(block: false, alt: "0")[$0$] cookies.

#math.equation(block: true, alt: "0 divided by 3 equals 0")[$0 ÷ 3 = 0$]Remember that we can always check division with the related multiplication fact. So, we know that

#math.equation(block: true, alt: "0 divided by 3 equals 0 because 0 times 3 equals 0 .")[$0 ÷ 3 = 0 #h(0.2em) "because" #h(0.2em) 0 · 3 = 0 .$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division of Zero]

For any real number #math.equation(block: false, alt: "a ,")[$a ,$] except #math.equation(block: false, alt: "0 , the fraction 0 over a equals 0")[$0 , frac(0, a) = 0$] and #math.equation(block: false, alt: "0 divided by a equals 0 .")[$0 ÷ a = 0 .$]
]

Zero divided by any real number except zero is zero.

#examplebox("Example 5")[][
Simplify: ⓐ #math.equation(block: false, alt: "0 divided by 5")[$#h(0.2em) 0 ÷ 5 #h(0.2em)$] ⓑ #math.equation(block: false, alt: "the fraction 0 over −2")[$frac(0, −2)$] ⓒ #math.equation(block: false, alt: "0 divided by the fraction 7 over 8")[$#h(0.2em) 0 ÷ frac(7, 8)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "0 divided by 5")[$0 ÷ 5$]],
  [Zero divided by any real number, except 0, is zero.], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "the fraction 0 over −2")[$frac(0, −2)$]],
  [Zero divided by any real number, except 0, is zero.], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "0 divided by the fraction 7 over 8")[$0 ÷ frac(7, 8)$]],
  [Zero divided by any real number, except 0, is zero.], [#math.equation(block: false, alt: "0")[$0$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "0 divided by 11")[$#h(0.2em) 0 ÷ 11$] ⓑ #math.equation(block: false, alt: "the fraction 0 over −6")[$#h(0.2em) frac(0, −6)$] ⓒ #math.equation(block: false, alt: "0 divided by the fraction 3 over 10")[$#h(0.2em) 0 ÷ frac(3, 10)$].

#solutionbox[
+ ⓐ 0
+ ⓑ 0
+ ⓒ 0
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "0 divided by the fraction 8 over 3")[$#h(0.2em) 0 ÷ frac(8, 3)$] ⓑ #math.equation(block: false, alt: "0 divided by open parenthesis −10 close parenthesis")[$#h(0.2em) 0 ÷ ( −10 )$] ⓒ #math.equation(block: false, alt: "0 divided by 12.75")[$#h(0.2em) 0 ÷ 12.75$].

#solutionbox[
+ ⓐ 0
+ ⓑ 0
+ ⓒ 0
]
]

Now let’s think about dividing a number #emph[by] zero. What is the result of dividing #math.equation(block: false, alt: "4")[$4$] by #math.equation(block: false, alt: "0 ?")[$0 ?$] Think about the related multiplication fact. Is there a number that multiplied by #math.equation(block: false, alt: "0")[$0$] gives #math.equation(block: false, alt: "4 ?")[$4 ?$]

#math.equation(block: true, alt: "4 divided by 0 equals ___ means ___ times 0 equals 4")[$4 ÷ 0 = \_\_\_ #h(0.2em) "means" #h(0.2em) \_\_\_ · 0 = 4$]Since any real number multiplied by #math.equation(block: false, alt: "0")[$0$] equals #math.equation(block: false, alt: "0 ,")[$0 ,$] there is no real number that can be multiplied by #math.equation(block: false, alt: "0")[$0$] to obtain #math.equation(block: false, alt: "4 .")[$4 .$] We can conclude that there is no answer to #math.equation(block: false, alt: "4 divided by 0 ,")[$4 ÷ 0 ,$] and so we say that division by zero is undefined.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division by Zero]

For any real number #math.equation(block: false, alt: "a , the fraction a over 0 ,")[$a , #h(0.2em) frac(a, 0) ,$] and #math.equation(block: false, alt: "a divided by 0")[$a ÷ 0$] are undefined.
]

Division #emph[by] zero is undefined.

#examplebox("Example 6")[][
Simplify: ⓐ #math.equation(block: false, alt: "7.5 divided by 0")[$#h(0.2em) 7.5 ÷ 0$] ⓑ #math.equation(block: false, alt: "the fraction −32 over 0")[$#h(0.2em) frac(−32, 0)$] ⓒ #math.equation(block: false, alt: "the fraction 4 over 9 divided by 0")[$#h(0.2em) frac(4, 9) ÷ 0$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "7.5 divided by 0")[$7.5 ÷ 0$]],
  [Division by zero is undefined.], [undefined],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "the fraction −32 over 0")[$frac(−32, 0)$]],
  [Division by zero is undefined.], [undefined],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "the fraction 4 over 9 divided by 0")[$frac(4, 9) ÷ 0$]],
  [Division by zero is undefined.], [undefined],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "16.4 divided by 0")[$#h(0.2em) 16.4 ÷ 0$] ⓑ #math.equation(block: false, alt: "the fraction −2 over 0")[$#h(0.2em) frac(−2, 0)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 5 divided by 0")[$#h(0.2em) frac(1, 5) ÷ 0$].

#solutionbox[
+ ⓐ undefined
+ ⓑ undefined
+ ⓒ undefined
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction −5 over 0")[$#h(0.2em) frac(−5, 0)$] ⓑ #math.equation(block: false, alt: "96.9 divided by 0")[$#h(0.2em) 96.9 ÷ 0$] ⓒ #math.equation(block: false, alt: "the fraction 4 over 15 divided by 0")[$#h(0.2em) frac(4, 15) ÷ 0$]

#solutionbox[
+ ⓐ undefined
+ ⓑ undefined
+ ⓒ undefined
]
]

We summarize the properties of zero.

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

#strong[Multiplication by Zero:] For any real number #math.equation(block: false, alt: "a ,")[$a ,$]

#math.equation(block: true, alt: "a times 0 equals 0 0 times a equals 0 The product of any number and 0 is 0.")[$#h(4em) a · 0 = 0 #h(2em) 0 · a = 0 #h(2em) "The product of any number and 0 is 0."$]

#strong[Division by Zero:] For any real number #math.equation(block: false, alt: "a , a not equal to 0")[$a , #h(0.2em) a ≠ 0$]

#math.equation(block: false, alt: "the fraction 0 over a equals 0")[$#h(4em) frac(0, a) = 0$] Zero divided by any real number, except itself, is zero.

#math.equation(block: false, alt: "the fraction a over 0")[$#h(4em) frac(a, 0)$] is undefined. Division by zero is undefined.
]


=== Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "3 x plus 15 minus 3 x .")[$3 x + 15 − 3 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 x plus 15 minus 3 x")[$3 x + 15 − 3 x$]]),
  [Notice the additive inverses, #math.equation(block: false, alt: "3 x")[$3 x$] and #math.equation(block: false, alt: "−3 x")[$−3 x$].], [#math.equation(block: false, alt: "0 plus 15")[$0 + 15$]],
  [Add.], [#math.equation(block: false, alt: "15")[$15$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−12 z plus 9 plus 12 z .")[$−12 z + 9 + 12 z .$]

#solutionbox[
9
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−25 u minus 18 plus 25 u .")[$−25 u − 18 + 25 u .$]

#solutionbox[
−18
]
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "4 open parenthesis 0.25 q close parenthesis .")[$4 ( 0.25 q ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 open parenthesis 0.25 q close parenthesis")[$4 ( 0.25 q )$]]),
  [Regroup, using the associative property.], [#math.equation(block: false, alt: "[ 4 open parenthesis 0.25 close parenthesis ] q")[$[ 4 ( 0.25 ) ] q$]],
  [Multiply.], [#math.equation(block: false, alt: "1.00 q")[$1.00 q$]],
  [Simplify; 1 is the multiplicative identity.], [#math.equation(block: false, alt: "q")[$q$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "2 open parenthesis 0.5 p close parenthesis .")[$2 ( 0.5 p ) .$]

#solutionbox[
#emph[p]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "25 open parenthesis 0.04 r close parenthesis .")[$25 ( 0.04 r ) .$]

#solutionbox[
#emph[r]
]
]

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the fraction 0 over n plus 5")[$frac(0, n + 5)$], where #math.equation(block: false, alt: "n not equal to −5")[$n ≠ −5$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 0 over n plus 5")[$frac(0, n + 5)$]]),
  [Zero divided by any real number except itself is zero.], [#math.equation(block: false, alt: "0")[$0$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 0 over m plus 7")[$frac(0, m + 7)$], where #math.equation(block: false, alt: "m not equal to −7")[$m ≠ −7$].

#solutionbox[
0
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 0 over d minus 4")[$frac(0, d − 4)$], where #math.equation(block: false, alt: "d not equal to 4")[$d ≠ 4$].

#solutionbox[
0
]
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "the fraction 10 minus 3 p over 0 .")[$frac(10 − 3 p, 0) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 10 minus 3 p over 0")[$frac(10 − 3 p, 0)$]]),
  [Division by zero is undefined.], [undefined],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 18 minus 6 c over 0 .")[$frac(18 − 6 c, 0) .$]

#solutionbox[
undefined
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 15 minus 4 q over 0 .")[$frac(15 − 4 q, 0) .$]

#solutionbox[
undefined
]
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 4 over 3 open parenthesis 6 x plus 12 close parenthesis .")[$frac(3, 4) · frac(4, 3) ( 6 x + 12 ) .$]

#solutionbox[
We cannot combine the terms in parentheses, so we multiply the two fractions first.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 3 over 4 times the fraction 4 over 3 open parenthesis 6 x plus 12 close parenthesis")[$frac(3, 4) · frac(4, 3) ( 6 x + 12 )$]]),
  [Multiply; the product of reciprocals is 1.], [#math.equation(block: false, alt: "1 open parenthesis 6 x plus 12 close parenthesis")[$1 ( 6 x + 12 )$]],
  [Simplify by recognizing the multiplicative identity.], [#math.equation(block: false, alt: "6 x plus 12")[$6 x + 12$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 5 times the fraction 5 over 2 open parenthesis 20 y plus 50 close parenthesis .")[$frac(2, 5) · frac(5, 2) ( 20 y + 50 ) .$]

#solutionbox[
20#emph[y] + 50
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 over 8 times the fraction 8 over 3 open parenthesis 12 z plus 16 close parenthesis .")[$frac(3, 8) · frac(8, 3) ( 12 z + 16 ) .$]

#solutionbox[
12#emph[z] + 16
]
]

All the properties of real numbers we have used in this chapter are summarized in .

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Property], [Of Addition], [Of Multiplication],
  [#strong[Commutative Property]], [], [],
  [If #emph[a] and #emph[b] are real numbers then…], [#math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$]], [#math.equation(block: false, alt: "a times b equals b times a")[$a · b = b · a$]],
  [#strong[Associative Property]], [], [],
  [If #emph[a], #emph[b], and #emph[c] are real numbers then…], [#math.equation(block: false, alt: "open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis")[$( a + b ) + c = a + ( b + c )$]], [#math.equation(block: false, alt: "open parenthesis a times b close parenthesis times c equals a times open parenthesis b times c close parenthesis")[$( a · b ) · c = a · ( b · c )$]],
  [#strong[Identity Property]], [#math.equation(block: false, alt: "0")[$0$] is the #strong[additive identity]], [#math.equation(block: false, alt: "1")[$1$] is the #strong[multiplicative identity]],
  [For any real number #emph[a],], [#math.equation(block: false, alt: "a plus 0 equals a; 0 plus a equals a")[$a + 0 = a \ 0 + a = a$]], [#math.equation(block: false, alt: "a times 1 equals a; 1 times a equals a")[$a · 1 = a \ 1 · a = a$]],
  [#strong[Inverse Property]], [#math.equation(block: false, alt: "minus a")[$− "a"$]is the #strong[additive inverse] of #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a , a not equal to 0")[$a , a ≠ 0$] #linebreak() #math.equation(block: false, alt: "1 / a")[$1 / "a"$] is the #strong[multiplicative inverse] of #math.equation(block: false, alt: "a")[$a$]],
  [For any real number #emph[a],], [#math.equation(block: false, alt: "a plus ( − a ) equals 0")[$a + "(" "−" "a" ")" #h(0.2em) = 0$]], [#math.equation(block: false, alt: "a times the fraction 1 over a equals 1")[$a · frac(1, a) = 1$]],
  [#strong[#strong[Distributive Property]] #linebreak() $#h(10em)$If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are real numbers, then #math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$]], [], [],
  [#strong[Properties of Zero]], [], [],
  [For any real number #emph[a], #linebreak()], [#math.equation(block: false, alt: "a times 0 equals 0; 0 times a equals 0")[$a ⋅ 0 = 0 \ 0 ⋅ a = 0$]], [],
  [For any real number #math.equation(block: false, alt: "a , a not equal to 0")[$a , a ≠ 0$]], [#math.equation(block: false, alt: "the fraction 0 over a equals 0")[$frac(0, a) = 0$] #linebreak() $$#math.equation(block: false, alt: "the fraction a over 0")[$frac(a, 0)$] is undefined], [],
))

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

- #link("https://www.openstax.org/l/24MultDivZero")[Multiplying and Dividing Involving Zero]
]


=== Key Concepts

- #emph[Identity Properties]- #emph[Identity Property of Addition:]
For any real number #emph[a]:  #math.equation(block: false, alt: "a plus 0 equals a 0 plus a equals a")[$a + 0 = a #h(1.5em) 0 + a = a #h(1.5em)$] #emph[0] is the #emph[additive identity]
- #emph[Identity Property of Multiplication:]
For any real number #emph[a]:  #math.equation(block: false, alt: "a times 1 equals a 1 times a equals a")[$a ⋅ 1 = a #h(1.5em) 1 ⋅ a = a #h(1.5em)$] #emph[1] is the #emph[multiplicative identity]
- #emph[Inverse Properties]- #emph[Inverse Property of Addition:] For any real number #emph[a]: #math.equation(block: false, alt: "a plus open parenthesis minus a close parenthesis equals 0 minus a")[$a + ( − a ) = 0 #h(1.5em) − a$] is the #emph[additive inverse] of #emph[a]
- #emph[Inverse Property of Multiplication:] For any real number #emph[a]: #math.equation(block: false, alt: "open parenthesis a not equal to 0 close parenthesis a times the fraction 1 over a equals 1 the fraction 1 over a")[$( a ≠ 0 ) #h(1.5em) a ⋅ frac(1, a) = 1 #h(1.5em) frac(1, a)$] is the #emph[multiplicative inverse] of #emph[a]
- #emph[Properties of Zero]- #emph[Multiplication by Zero:] For any real number #emph[a],
#math.equation(block: false, alt: "a times 0 equals 0, 0 times a equals 0, The product of any number and 0 is 0.")[$a ⋅ 0 = 0 &  &  & 0 ⋅ a = 0 &  &  & "The product of any number and 0 is 0."$]
- #emph[Division of Zero:] For any real number #emph[a],
#math.equation(block: false, alt: "the fraction 0 over a equals 0, Zero divided by any real number, except itself, is zero.")[$frac(0, a) = 0 &  &  &  &  &  & "Zero divided by any real number, except itself, is zero."$]
- #emph[Division by Zero:]For any real number #emph[a,] #math.equation(block: false, alt: "the fraction a over 0")[$frac(a, 0)$] is undefined and #math.equation(block: false, alt: "a divided by 0")[$a ÷ 0$] is undefined. Division by zero is undefined.


==== Practice Makes Perfect

#strong[Recognize the Identity Properties of Addition and Multiplication]

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

#math.equation(block: true, alt: "101 plus 0 equals 101")[$101 + 0 = 101$]

#math.equation(block: true, alt: "the fraction 3 over 5 open parenthesis 1 close parenthesis equals the fraction 3 over 5")[$frac(3, 5) ( 1 ) = frac(3, 5)$]

#solutionbox[
identity property of multiplication
]

#math.equation(block: true, alt: "−9 times 1 equals −9")[$−9 · 1 = −9$]

#math.equation(block: true, alt: "0 plus 64 equals 64")[$0 + 64 = 64$]

#solutionbox[
identity property of addition
]

#strong[Use the Inverse Properties of Addition and Multiplication]

In the following exercises, find the multiplicative inverse.

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

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

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 14")[$frac(1, 14)$]
]

#math.equation(block: true, alt: "−17")[$−17$]

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

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 19")[$− frac(1, 19)$]
]

#math.equation(block: true, alt: "the fraction 7 over 12")[$frac(7, 12)$]

#math.equation(block: true, alt: "the fraction 8 over 13")[$frac(8, 13)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 8")[$frac(13, 8)$]
]

#math.equation(block: true, alt: "minus the fraction 3 over 10")[$− frac(3, 10)$]

#math.equation(block: true, alt: "minus the fraction 5 over 12")[$− frac(5, 12)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 12 over 5")[$− frac(12, 5)$]
]

#math.equation(block: true, alt: "0.8")[$0.8$]

#math.equation(block: true, alt: "0.4")[$0.4$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$]
]

#math.equation(block: true, alt: "−0.2")[$−0.2$]

#math.equation(block: true, alt: "−0.5")[$−0.5$]

#solutionbox[
−2
]

#strong[Use the Properties of Zero]

In the following exercises, simplify using the properties of zero.

#math.equation(block: true, alt: "48 times 0")[$48 · 0$]

#math.equation(block: true, alt: "the fraction 0 over 6")[$frac(0, 6)$]

#solutionbox[
0
]

#math.equation(block: true, alt: "the fraction 3 over 0")[$frac(3, 0)$]

#math.equation(block: true, alt: "22 times 0")[$22 · 0$]

#solutionbox[
0
]

#math.equation(block: true, alt: "0 divided by the fraction 11 over 12")[$0 ÷ frac(11, 12)$]

#math.equation(block: true, alt: "the fraction 6 over 0")[$frac(6, 0)$]

#solutionbox[
undefined
]

#math.equation(block: true, alt: "the fraction 0 over 3")[$frac(0, 3)$]

#math.equation(block: true, alt: "0 divided by the fraction 7 over 15")[$0 ÷ frac(7, 15)$]

#solutionbox[
0
]

#math.equation(block: true, alt: "0 times the fraction 8 over 15")[$0 · frac(8, 15)$]

#math.equation(block: true, alt: "open parenthesis −3.14 close parenthesis open parenthesis 0 close parenthesis")[$( −3.14 ) ( 0 )$]

#solutionbox[
0
]

#math.equation(block: true, alt: "5.72 divided by 0")[$5.72 ÷ 0$]

#math.equation(block: true, alt: "the fraction the fraction 1 over 10 over 0")[$frac(frac(1, 10), 0)$]

#solutionbox[
undefined
]

#strong[Simplify Expressions using the Properties of Identities, Inverses, and Zero]

In the following exercises, simplify using the properties of identities, inverses, and zero.

#math.equation(block: true, alt: "19 a plus 44 minus 19 a")[$19 a + 44 − 19 a$]

#math.equation(block: true, alt: "27 c plus 16 minus 27 c")[$27 c + 16 − 27 c$]

#solutionbox[
16
]

#math.equation(block: true, alt: "38 plus 11 r minus 38")[$38 + 11 r − 38$]

#math.equation(block: true, alt: "92 plus 31 s minus 92")[$92 + 31 s − 92$]

#solutionbox[
31#emph[s]
]

#math.equation(block: true, alt: "10 open parenthesis 0.1 d close parenthesis")[$10 ( 0.1 d )$]

#math.equation(block: true, alt: "100 open parenthesis 0.01 p close parenthesis")[$100 ( 0.01 p )$]

#solutionbox[
#emph[p]
]

#math.equation(block: true, alt: "5 open parenthesis 0.6 q close parenthesis")[$5 ( 0.6 q )$]

#math.equation(block: true, alt: "40 open parenthesis 0.05 n close parenthesis")[$40 ( 0.05 n )$]

#solutionbox[
2#emph[n]
]

#math.equation(block: false, alt: "the fraction 0 over r plus 20")[$frac(0, r + 20)$], where #math.equation(block: false, alt: "r not equal to −20")[$r ≠ −20$]

#math.equation(block: false, alt: "the fraction 0 over s plus 13")[$frac(0, s + 13)$], where #math.equation(block: false, alt: "s not equal to −13")[$s ≠ −13$]

#solutionbox[
0
]

#math.equation(block: false, alt: "the fraction 0 over u minus 4.99")[$frac(0, u − 4.99)$], where #math.equation(block: false, alt: "u not equal to 4.99")[$u ≠ 4.99$]

#math.equation(block: false, alt: "the fraction 0 over v minus 65.1")[$frac(0, v − 65.1)$], where #math.equation(block: false, alt: "v not equal to 65.1")[$v ≠ 65.1$]

#solutionbox[
0
]

#math.equation(block: false, alt: "0 divided by open parenthesis x minus the fraction 1 over 2 close parenthesis")[$0 ÷ ( x − frac(1, 2) )$], where #math.equation(block: false, alt: "x not equal to the fraction 1 over 2")[$x ≠ frac(1, 2)$]

#math.equation(block: false, alt: "0 divided by open parenthesis y minus the fraction 1 over 6 close parenthesis")[$0 ÷ ( y − frac(1, 6) )$], where #math.equation(block: false, alt: "y not equal to the fraction 1 over 6")[$y ≠ frac(1, 6)$]

#solutionbox[
0
]

#math.equation(block: false, alt: "the fraction 32 minus 5 a over 0")[$frac(32 − 5 a, 0)$], where #math.equation(block: false, alt: "32 minus 5 a not equal to 0")[$32 − 5 a ≠ 0$]

#math.equation(block: false, alt: "the fraction 28 minus 9 b over 0")[$frac(28 − 9 b, 0)$], where #math.equation(block: false, alt: "28 minus 9 b not equal to 0")[$28 − 9 b ≠ 0$]

#solutionbox[
undefined
]

#math.equation(block: false, alt: "the fraction 2.1 plus 0.4 c over 0")[$frac(2.1 + 0.4 c, 0)$], where #math.equation(block: false, alt: "2.1 plus 0.4 c not equal to 0")[$2.1 + 0.4 c ≠ 0$]

#math.equation(block: false, alt: "the fraction 1.75 plus 9 f over 0")[$frac(1.75 + 9 f, 0)$], where #math.equation(block: false, alt: "1.75 plus 9 f not equal to 0")[$1.75 + 9 f ≠ 0$]

#solutionbox[
undefined
]

#math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 plus the fraction 9 over 10 m close parenthesis divided by 0")[$( frac(3, 4) + frac(9, 10) m ) ÷ 0$], where #math.equation(block: false, alt: "the fraction 3 over 4 plus the fraction 9 over 10 m not equal to 0")[$frac(3, 4) + frac(9, 10) m ≠ 0$]

#math.equation(block: false, alt: "open parenthesis the fraction 5 over 16 n minus the fraction 3 over 7 close parenthesis divided by 0")[$( frac(5, 16) n − frac(3, 7) ) ÷ 0$], where #math.equation(block: false, alt: "the fraction 5 over 16 n minus the fraction 3 over 7 not equal to 0")[$frac(5, 16) n − frac(3, 7) ≠ 0$]

#solutionbox[
undefined
]

#math.equation(block: true, alt: "the fraction 9 over 10 times the fraction 10 over 9 open parenthesis 18 p minus 21 close parenthesis")[$frac(9, 10) · frac(10, 9) ( 18 p − 21 )$]

#math.equation(block: true, alt: "the fraction 5 over 7 times the fraction 7 over 5 open parenthesis 20 q minus 35 close parenthesis")[$frac(5, 7) · frac(7, 5) ( 20 q − 35 )$]

#solutionbox[
20#emph[q] − 35
]

#math.equation(block: true, alt: "15 times the fraction 3 over 5 open parenthesis 4 d plus 10 close parenthesis")[$15 · frac(3, 5) ( 4 d + 10 )$]

#math.equation(block: true, alt: "18 times the fraction 5 over 6 open parenthesis 15 h plus 24 close parenthesis")[$18 · frac(5, 6) ( 15 h + 24 )$]

#solutionbox[
225#emph[h] + 360
]


==== Everyday Math

#strong[Insurance copayment] Carrie had to have #math.equation(block: false, alt: "5")[$5$] fillings done. Each filling cost #math.equation(block: false, alt: "$80 .")[$"$80" .$] Her dental insurance required her to pay #math.equation(block: false, alt: "20 %")[$20 %$] of the cost. Calculate Carrie’s cost

+ ⓐ by finding her copay for each filling, then finding her total cost for #math.equation(block: false, alt: "5")[$5$] fillings, and
+ ⓑ by multiplying #math.equation(block: false, alt: "5 open parenthesis 0.20 close parenthesis open parenthesis 80 close parenthesis .")[$5 ( 0.20 ) ( 80 ) .$]
+ ⓒ Which of the Properties of Real Numbers did you use for part (b)?

#strong[Cooking time] Helen bought a #math.equation(block: false, alt: "24-pound")[$"24-pound"$] turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in the oven. She wants to allow #math.equation(block: false, alt: "20")[$20$] minutes per pound cooking time.

+ ⓐ Calculate the length of time needed to roast the turkey by multiplying #math.equation(block: false, alt: "24 times 20")[$24 · 20$] to find the number of minutes and then multiplying the product by #math.equation(block: false, alt: "the fraction 1 over 60")[$frac(1, 60)$] to convert minutes into hours.
+ ⓑ Multiply #math.equation(block: false, alt: "24 open parenthesis 20 times the fraction 1 over 60 close parenthesis .")[$24 ( 20 · frac(1, 60) ) .$]
+ ⓒ Which of the Properties of Real Numbers allows you to multiply #math.equation(block: false, alt: "24 open parenthesis 20 times the fraction 1 over 60 close parenthesis")[$24 ( 20 · frac(1, 60) )$] instead of #math.equation(block: false, alt: "open parenthesis 24 times 20 close parenthesis the fraction 1 over 60 ?")[$( 24 · 20 ) frac(1, 60) ?$]

#solutionbox[
+ ⓐ 8 hours
+ ⓑ 8
+ ⓒ associative property of multiplication
]


==== Writing Exercises

In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.

How can the use of the properties of real numbers make it easier to simplify expressions?

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[Math self-assessment grid for properties of addition and multiplication (identity, inverse, zero). Learners can indicate if they understand 'Confidently', 'With some help', or 'No-I don't get it!'], alt: "Math self-assessment grid for properties of addition and multiplication (identity, inverse, zero). Learners can indicate if they understand 'Confidently', 'With some help', or 'No-I don't get it!'", 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?