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== 1.5#h(0.6em)Properties of Real Numbers

#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[Elementary Algebra 2e] chapter, Foundations.
]


=== Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add #math.equation(block: false, alt: "8 plus 9")[$8 + 9$] or #math.equation(block: false, alt: "9 plus 8 ,")[$9 + 8 ,$] the results are the same—they both equal 17. So, #math.equation(block: false, alt: "8 plus 9 equals 9 plus 8 .")[$8 + 9 = 9 + 8 .$] The order in which we add does not matter!

Similarly, when multiplying two numbers, the order does not affect the result. If we multiply #math.equation(block: false, alt: "9 times 8")[$9 · 8$] or #math.equation(block: false, alt: "8 times 9")[$8 · 9$] the results are the same—they both equal 72. So, #math.equation(block: false, alt: "9 times 8 equals 8 times 9 .")[$9 · 8 = 8 · 9 .$] The order in which we multiply does not matter!

These examples illustrate the #strong[Commutative Property].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Commutative Property]

#math.equation(block: true, alt: "of Addition, If a and b are real numbers, then, a plus b equals b plus a .; of Multiplication, If a and b are real numbers, then, a times b equals b times a .")[$"of Addition" &  &  & "If" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b #h(0.2em) "are real numbers, then" &  &  & a + b = b + a . \ "of Multiplication" &  &  & "If" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b #h(0.2em) "are real numbers, then" &  &  & a · b = b · a .$]When adding or multiplying, changing the #emph[order] gives the same result.
]

The Commutative Property has to do with order. We subtract #math.equation(block: false, alt: "9 minus 8")[$9 − 8$] and #math.equation(block: false, alt: "8 minus 9")[$8 − 9$], and see that #math.equation(block: false, alt: "9 minus 8 not equal to 8 minus 9 .")[$9 − 8 ≠ 8 − 9 .$] Since changing the order of the subtraction does not give the same result, we know that #emph[subtraction is not commutative].

#emph[Division is not commutative either]. Since #math.equation(block: false, alt: "12 divided by 3 not equal to 3 divided by 12 ,")[$12 ÷ 3 ≠ 3 ÷ 12 ,$] changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!

Addition and multiplication #emph[are] commutative.

Subtraction and division are #emph[not] commutative.

When adding three numbers, changing the grouping of the numbers gives the same result. For example, #math.equation(block: false, alt: "open parenthesis 7 plus 8 close parenthesis plus 2 equals 7 plus open parenthesis 8 plus 2 close parenthesis ,")[$( 7 + 8 ) + 2 = 7 + ( 8 + 2 ) ,$] since each side of the equation equals 17.

This is true for multiplication, too. For example, #math.equation(block: false, alt: "open parenthesis 5 times the fraction 1 over 3 close parenthesis times 3 equals 5 times open parenthesis the fraction 1 over 3 times 3 close parenthesis ,")[$( 5 · frac(1, 3) ) · 3 = 5 · ( frac(1, 3) · 3 ) ,$] since each side of the equation equals 5.

These examples illustrate the #strong[Associative Property].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Associative Property]

#math.equation(block: true, alt: "of Addition, If a , b , and c are real numbers, then, open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis .; of Multiplication, If a , b , and c are real numbers, then, open parenthesis a times b close parenthesis times c equals a times open parenthesis b times c close parenthesis .")[$"of Addition" &  &  & "If" #h(0.2em) a , b , "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & ( a + b ) + c = a + ( b + c ) . \ "of Multiplication" &  &  & "If" #h(0.2em) "a" , b , "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & ( a · b ) · c = a · ( b · c ) .$]When adding or multiplying, changing the #emph[grouping] gives the same result.
]

The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

#math.equation(block: true, alt: "open parenthesis 10 minus 3 close parenthesis minus 2 not equal to 10 minus open parenthesis 3 minus 2 close parenthesis, open parenthesis 24 divided by 4 close parenthesis divided by 2 not equal to 24 divided by open parenthesis 4 divided by 2 close parenthesis; 7 minus 2 not equal to 10 minus 1, 6 divided by 2 not equal to 24 divided by 2; 5 not equal to 9, 3 not equal to 12")[$( 10 − 3 ) − 2 ≠ 10 − ( 3 − 2 ) &  &  & ( 24 ÷ 4 ) ÷ 2 ≠ 24 ÷ ( 4 ÷ 2 ) \ #h(0.5em) 7 − 2 ≠ 10 − 1 &  &  & #h(0.5em) 6 ÷ 2 ≠ 24 ÷ 2 \ #h(0.1em) 5 ≠ 9 &  &  & #h(0.6em) 3 ≠ 12$]When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.

#examplebox("Example 1")[][
Simplify: #math.equation(block: false, alt: "18 p plus 6 q plus 15 p plus 5 q .")[$18 p + 6 q + 15 p + 5 q .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "18 p plus 6 q plus 15 p plus 5 q")[$18 p + 6 q + 15 p + 5 q$]]),
  [Use the Commutative Property of addition to reorder so that like terms are together.], [#math.equation(block: false, alt: "18 p plus 15 p plus 6 q plus 5 q")[$18 p + 15 p + 6 q + 5 q$]],
  [Add like terms.], [#math.equation(block: false, alt: "33 p plus 11 q")[$33 p + 11 q$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "23 r plus 14 s plus 9 r plus 15 s .")[$23 r + 14 s + 9 r + 15 s .$]

#solutionbox[
#math.equation(block: true, alt: "32 r plus 29 s")[$32 r + 29 s$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:#math.equation(block: false, alt: "37 m plus 21 n plus 4 m minus 15 n .")[$37 m + 21 n + 4 m − 15 n .$]

#solutionbox[
#math.equation(block: true, alt: "41 m plus 6 n")[$41 m + 6 n$]
]
]

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 5 over 13 plus the fraction 3 over 4 close parenthesis plus the fraction 1 over 4 .")[$( frac(5, 13) + frac(3, 4) ) + frac(1, 4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 5 over 13 plus the fraction 3 over 4 close parenthesis plus the fraction 1 over 4")[$( frac(5, 13) + frac(3, 4) ) + frac(1, 4)$]]),
  [Notice that the last 2 terms have a common denominator, so change the grouping.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus open parenthesis the fraction 3 over 4 plus the fraction 1 over 4 close parenthesis")[$frac(5, 13) + ( frac(3, 4) + frac(1, 4) )$]],
  [Add in parentheses first.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus open parenthesis the fraction 4 over 4 close parenthesis")[$frac(5, 13) + ( frac(4, 4) )$]],
  [Simplify the fraction.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus 1")[$frac(5, 13) + 1$]],
  [Add.], [#math.equation(block: false, alt: "1 the fraction 5 over 13")[$1 frac(5, 13)$]],
  [Convert to an improper fraction.], [#math.equation(block: false, alt: "the fraction 18 over 13")[$frac(18, 13)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 7 over 15 plus the fraction 5 over 8 close parenthesis plus the fraction 3 over 8 .")[$( frac(7, 15) + frac(5, 8) ) + frac(3, 8) .$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 7 over 15")[$1 frac(7, 15)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 2 over 9 plus the fraction 7 over 12 close parenthesis plus the fraction 5 over 12 .")[$( frac(2, 9) + frac(7, 12) ) + frac(5, 12) .$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 2 over 9")[$1 frac(2, 9)$]
]
]


=== Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the #strong[additive identity]. The #strong[Identity Property of Addition] that states that for any real number #math.equation(block: false, alt: "a , a plus 0 equals a")[$a , a + 0 = a$] and #math.equation(block: false, alt: "0 plus a equals a .")[$0 + a = a .$]

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the #strong[multiplicative identity]. The #strong[Identity Property of Multiplication] that states that for any real number #math.equation(block: false, alt: "a , a times 1 equals a")[$a , a · 1 = a$] and #math.equation(block: false, alt: "1 times a equals a .")[$1 · a = a .$]

We summarize the Identity Properties here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Identity Property]

#math.equation(block: true, alt: "of Addition For any real number a : a plus 0 equals a, 0 plus a equals a; 0 is the additive identity; of Multiplication For any real number a : a times 1 equals a, 1 times a equals a; 1 is the multiplicative identity")[$"of Addition" #h(1em) "For any real number" #h(0.2em) a : #h(0.2em) a + 0 = a &  &  & 0 + a = a \ #h(1.5em) 0 #h(0.2em) "is the" #h(0.2em) "additive identity" &  &  &  \ "of Multiplication" #h(1em) "For any real number" #h(0.2em) a : #h(0.2em) a · 1 = a &  &  & 1 · a = a \ #h(1.5em) 1 #h(0.2em) "is the" #h(0.2em) "multiplicative identity"$]
]

What number added to 5 gives the additive identity, 0? We know

#figure(figph[Figure shows the expression 5 plus open parentheses minus 5 close parentheses equals 0.], alt: "Figure shows the expression 5 plus open parentheses minus 5 close parentheses equals 0.", caption: none)

The missing number was the opposite of the number!

We call #math.equation(block: false, alt: "− a")[$"−" a$] the #strong[additive inverse] of #math.equation(block: false, alt: "a .")[$a .$] #emph[The opposite of a number is its additive inverse.] A number and its opposite add to zero, which is the additive identity. This leads to the #strong[Inverse Property of Addition] that states for any real number #math.equation(block: false, alt: "a , a plus open parenthesis − a close parenthesis equals 0 .")[$a , a + ( "−" a ) = 0 .$]

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

#figure(figph[2 by 3 times 3 by 2 equals 1.], alt: "2 by 3 times 3 by 2 equals 1.", caption: none)

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 #emph[a]. #emph[The reciprocal of a number is its multiplicative inverse.] This leads to the #strong[Inverse Property of Multiplication] that states that for any real number #math.equation(block: false, alt: "a , a not equal to 0 , a times the fraction 1 over a equals 1 .")[$a , a ≠ 0 , a · frac(1, a) = 1 .$]

We’ll formally state the inverse properties here.

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

#math.equation(block: true, alt: "of Addition, For any real number a , a plus open parenthesis − a close parenthesis equals 0; − a is the additive inverse of a; A number and its o p p o s i t e add to zero.; of Multiplication, For any real number a , a not equal to 0 , a times the fraction 1 over a equals 1 .; the fraction 1 over a is the multiplicative inverse of a .; A number and its r e c i p r o c a l multiply to one.")[$"of Addition" &  &  & "For any real number" #h(0.2em) a , #h(2em) a + ( "−" a ) = 0 \  &  &  & "−" a #h(0.2em) "is the" #h(0.2em) "additive inverse" #h(0.2em) "of" #h(0.2em) a \  &  &  & "A number and its" #h(0.2em) o p p o s i t e #h(0.2em) "add to zero." \  \  \  \  \ "of Multiplication" &  &  & "For any real number" #h(0.2em) a , #h(2em) a ≠ 0 , #h(0.5em) a · frac(1, a) = 1 . \  &  &  & frac(1, a) #h(0.2em) "is the" #h(0.2em) "multiplicative inverse" #h(0.2em) "of" #h(0.2em) a . \  &  &  & "A number and its" #h(0.2em) r e c i p r o c a l #h(0.2em) "multiply to one."$]
]

The Identity Property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

What about division involving zero? What is #math.equation(block: false, alt: "0 divided by 3 ?")[$0 ÷ 3 ?$] Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So, #math.equation(block: false, alt: "0 divided by 3 equals 0 .")[$0 ÷ 3 = 0 .$]

We can check division with the related multiplication fact. So we know #math.equation(block: false, alt: "0 divided by 3 equals 0")[$0 ÷ 3 = 0$] because #math.equation(block: false, alt: "0 times 3 equals 0 .")[$0 · 3 = 0 .$]

Now think about dividing #emph[by] zero. What is the result of dividing 4 by #math.equation(block: false, alt: "0 ?")[$0 ?$] Think about the related multiplication fact:

#figure(figph[4 divided by 0 equals question mark means question mark times 0 equals 4.], alt: "4 divided by 0 equals question mark means question mark times 0 equals 4.", caption: none)

Is there a number that multiplied by 0 gives #math.equation(block: false, alt: "4 ?")[$4 ?$] Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4. We 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 0 is #strong[undefined].

We summarize the properties of zero here.

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

Multiplication by Zero: For any real number #emph[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.")[$#h(6.5em) a · 0 = 0 #h(1em) 0 · a = 0 #h(2em) "The product of any number and 0 is 0."$]

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

#math.equation(block: true, alt: "the fraction 0 over a equals 0, Zero divided by any real number, except itself, is zero.; the fraction a over 0 is undefined, Division by zero is undefined.")[$frac(0, a) = 0 &  &  & #h(0.5em) "Zero divided by any real number, except itself, is zero." \ frac(a, 0) #h(0.2em) "is undefined" &  &  & #h(0.5em) "Division by zero is undefined."$]
]

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

#examplebox("Example 3")[][
Simplify: #math.equation(block: false, alt: "−84 n plus open parenthesis −73 n close parenthesis plus 84 n .")[$−84 n + ( −73 n ) + 84 n .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−84 n plus open parenthesis −73 n close parenthesis plus 84 n")[$−84 n + ( −73 n ) + 84 n$]]),
  [Notice that the first and third terms are opposites; use the Commutative Property of addition to re-order the terms.], [#math.equation(block: false, alt: "−84 n plus 84 n plus open parenthesis −73 n close parenthesis")[$−84 n + 84 n + ( −73 n )$]],
  [Add left to right.], [#math.equation(block: false, alt: "0 plus open parenthesis −73 n close parenthesis")[$0 + ( −73 n )$]],
  [Add.], [#math.equation(block: false, alt: "−73 n")[$−73 n$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−27 a plus open parenthesis −48 a close parenthesis plus 27 a .")[$−27 a + ( −48 a ) + 27 a .$]

#solutionbox[
#math.equation(block: true, alt: "−48 a")[$−48 a$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "39 x plus open parenthesis −92 x close parenthesis plus open parenthesis −39 x close parenthesis .")[$39 x + ( −92 x ) + ( −39 x ) .$]

#solutionbox[
#math.equation(block: true, alt: "−92 x")[$−92 x$]
]
]

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 8 over 23 times the fraction 15 over 7 .")[$frac(7, 15) · frac(8, 23) · frac(15, 7) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 8 over 23 times the fraction 15 over 7")[$frac(7, 15) · frac(8, 23) · frac(15, 7)$]]),
  [Notice the first and third terms are reciprocals, so use the Commutative Property of multiplication to re-order the factors.], [#math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 15 over 7 times the fraction 8 over 23")[$frac(7, 15) · frac(15, 7) · frac(8, 23)$]],
  [Multiply left to right.], [#math.equation(block: false, alt: "1 times the fraction 8 over 23")[$1 · frac(8, 23)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 8 over 23")[$frac(8, 23)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 9 over 16 times the fraction 5 over 49 times the fraction 16 over 9 .")[$frac(9, 16) · frac(5, 49) · frac(16, 9) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 49")[$frac(5, 49)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 6 over 17 times the fraction 11 over 25 times the fraction 17 over 6 .")[$frac(6, 17) · frac(11, 25) · frac(17, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 25")[$frac(11, 25)$]
]
]

The next example makes us aware of the distinction between dividing 0 by some number or some number being divided by 0.

#examplebox("Example 5")[][
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")[$n ≠ "−" 5$] ⓑ #math.equation(block: false, alt: "the fraction 10 minus 3 p over 0 ,")[$frac(10 − 3 p, 0) ,$] where #math.equation(block: false, alt: "10 minus 3 p not equal to 0 .")[$10 − 3 p ≠ 0 .$]

#solutionbox[
ⓐ  #linebreak() 
#math.equation(block: false, alt: "the fraction 0 over n plus 5; Zero divided by any real number except itself is 0., 0")[$&  &  & #h(4em) frac(0, n + 5) \ "Zero divided by any real number except itself is 0." &  &  & #h(4em) 0$]

ⓑ  #linebreak() 
#math.equation(block: false, alt: "the fraction 10 minus 3 p over 0; Division by 0 is undefined., undefined")[$&  &  & #h(12em) frac(10 − 3 p, 0) \ "Division by 0 is undefined." &  &  & #h(12em) "undefined"$]
]
]

#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")[$m ≠ "−" 7$] ⓑ #math.equation(block: false, alt: "the fraction 18 minus 6 c over 0 ,")[$frac(18 − 6 c, 0) ,$] where #math.equation(block: false, alt: "18 minus 6 c not equal to 0 .")[$18 − 6 c ≠ 0 .$]

#solutionbox[
ⓐ 0 ⓑ undefined
]
]

#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")[$d ≠ 4$] ⓑ #math.equation(block: false, alt: "the fraction 15 minus 4 q over 0 ,")[$frac(15 − 4 q, 0) ,$] where #math.equation(block: false, alt: "15 minus 4 q not equal to 0 .")[$15 − 4 q ≠ 0 .$]

#solutionbox[
ⓐ 0 ⓑ undefined
]
]


=== Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need \$9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times \$9 so \$27 and 3 times 1 quarter, so 75 cents. In total, they need \$27.75. If you think about doing the math in this way, you are using the Distributive Property.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Distributive Property]

#math.equation(block: true, alt: "If a , b , and c are real numbers, then, a open parenthesis b plus c close parenthesis equals a b plus a c; open parenthesis b plus c close parenthesis a equals b a plus c a; a open parenthesis b minus c close parenthesis equals a b minus a c; open parenthesis b minus c close parenthesis a equals b a minus c a")[$"If" #h(0.2em) a , b , "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & a ( b + c ) = a b + a c \  &  &  & ( b + c ) a = b a + c a \  &  &  & a ( b − c ) = a b − a c \  &  &  & ( b − c ) a = b a − c a$]
]

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "3 open parenthesis x plus 4 close parenthesis .")[$3 ( x + 4 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 open parenthesis x plus 4 close parenthesis")[$3 ( x + 4 )$]]),
  [Distribute.], [#math.equation(block: false, alt: "3 times x plus 3 times 4")[$3 · x + 3 · 4$]],
  [Multiply.], [#math.equation(block: false, alt: "3 x plus 12")[$3 x + 12$]],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "4 x plus 8")[$4 x + 8$]
]
]

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

#solutionbox[
#math.equation(block: true, alt: "6 x plus 42")[$6 x + 42$]
]
]

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in  would look like this:

#figure(figph[The expression is 3 open parentheses x plus 4 close parentheses. Two arrows originate from 3. One points to x, the other to 4.], alt: "The expression is 3 open parentheses x plus 4 close parentheses. Two arrows originate from 3. One points to x, the other to 4.", caption: none)

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "8 open parenthesis the fraction 3 over 8 x plus the fraction 1 over 4 close parenthesis .")[$8 ( frac(3, 8) x + frac(1, 4) ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The distributive property is applied to the expression 8(3/8x + 1/4), showing 8 multiplied by each term inside the parentheses.], alt: "The distributive property is applied to the expression 8(3/8x + 1/4), showing 8 multiplied by each term inside the parentheses.", caption: none)]),
  [Distribute.], [#figure(figph[The mathematical expression 8 times 3 over 8x plus 8 times 1 over 4 is shown on a white background.], alt: "The mathematical expression 8 times 3 over 8x plus 8 times 1 over 4 is shown on a white background.", caption: none)],
  [Multiply.], [#figure(figph[The mathematical expression '3x + 2' is displayed in black text against a white background.], alt: "The mathematical expression '3x + 2' is displayed in black text against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "6 open parenthesis the fraction 5 over 6 y plus the fraction 1 over 2 close parenthesis .")[$6 ( frac(5, 6) y + frac(1, 2) ) .$]

#solutionbox[
#math.equation(block: true, alt: "5 y plus 3")[$5 y + 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "12 open parenthesis the fraction 1 over 3 n plus the fraction 3 over 4 close parenthesis .")[$12 ( frac(1, 3) n + frac(3, 4) ) .$]

#solutionbox[
#math.equation(block: true, alt: "4 n plus 9")[$4 n + 9$]
]
]

Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "100 open parenthesis 0.3 plus 0.25 q close parenthesis .")[$100 ( 0.3 + 0.25 q ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays the distributive property, showing 100 multiplied by (0.3 + 0.25q). Blue arrows indicate that 100 distributes to both 0.3 and 0.25q within the parenthesis.], alt: "A mathematical expression displays the distributive property, showing 100 multiplied by (0.3 + 0.25q). Blue arrows indicate that 100 distributes to both 0.3 and 0.25q within the parenthesis.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression showing the sum of two products: 100 multiplied by 0.3, and 100 multiplied by 0.25q.], alt: "A mathematical expression showing the sum of two products: 100 multiplied by 0.3, and 100 multiplied by 0.25q.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression '30 + 25q' is displayed on a white background. It represents an algebraic equation with constants and a variable 'q'.], alt: "A mathematical expression '30 + 25q' is displayed on a white background. It represents an algebraic equation with constants and a variable 'q'.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "100 open parenthesis 0.7 plus 0.15 p close parenthesis .")[$100 ( 0.7 + 0.15 p ) .$]

#solutionbox[
#math.equation(block: true, alt: "70 plus 15 p")[$70 + 15 p$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "100 open parenthesis 0.04 plus 0.35 d close parenthesis .")[$100 ( 0.04 + 0.35 d ) .$]

#solutionbox[
#math.equation(block: true, alt: "4 plus 35 d")[$4 + 35 d$]
]
]

When we distribute a negative number, we need to be extra careful to get the signs correct!

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "−11 open parenthesis 4 minus 3 a close parenthesis .")[$−11 ( 4 − 3 a ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−11 open parenthesis 4 minus 3 a close parenthesis")[$−11 ( 4 − 3 a )$]]),
  [Distribute.], [#math.equation(block: false, alt: "−11 times 4 minus open parenthesis −11 close parenthesis times 3 a")[$−11 · 4 − ( −11 ) · 3 a$]],
  [Multiply.], [#math.equation(block: false, alt: "−44 minus open parenthesis −33 a close parenthesis")[$−44 − ( −33 a )$]],
  [Simplify.], [#math.equation(block: false, alt: "−44 plus 33 a")[$−44 + 33 a$]],
))

Notice that you could also write the result as #math.equation(block: false, alt: "33 a minus 44 .")[$33 a − 44 .$] Do you know why?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−5 open parenthesis 2 minus 3 a close parenthesis .")[$−5 ( 2 − 3 a ) .$]

#solutionbox[
#math.equation(block: true, alt: "−10 plus 15 a")[$−10 + 15 a$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−7 open parenthesis 8 minus 15 y close parenthesis .")[$−7 ( 8 − 15 y ) .$]

#solutionbox[
#math.equation(block: true, alt: "−56 plus 105 y")[$−56 + 105 y$]
]
]

In the next example, we will show how to use the #strong[Distributive Property] to find the opposite of an expression.

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "− open parenthesis y plus 5 close parenthesis .")[$"−" ( y + 5 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− open parenthesis y plus 5 close parenthesis")[$"−" ( y + 5 )$]]),
  [Multiplying by #math.equation(block: false, alt: "−1")[$−1$] results in the opposite.], [#math.equation(block: false, alt: "−1 open parenthesis y plus 5 close parenthesis")[$−1 ( y + 5 )$]],
  [Distribute.], [#math.equation(block: false, alt: "−1 times y plus open parenthesis −1 close parenthesis times 5")[$−1 · y + ( −1 ) · 5$]],
  [Simplify.], [#math.equation(block: false, alt: "minus y plus open parenthesis −5 close parenthesis")[$− y + ( −5 )$]],
  [Simplify.], [#math.equation(block: false, alt: "minus y minus 5")[$− y − 5$]],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "− z plus 11")[$"−" z + 11$]
]
]

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

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

There will be times when we’ll need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the Distributive Property, which removes the parentheses. The next two examples will illustrate this.

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "8 minus 2 open parenthesis x plus 3 close parenthesis")[$8 − 2 ( x + 3 )$]

#solutionbox[
We follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "8 minus 2 open parenthesis x plus 3 close parenthesis")[$8 − 2 ( x + 3 )$]]),
  [Distribute.], [#math.equation(block: false, alt: "8 minus 2 times x minus 2 times 3")[$8 − 2 · x − 2 · 3$]],
  [Multiply.], [#math.equation(block: false, alt: "8 minus 2 x minus 6")[$8 − 2 x − 6$]],
  [Combine like terms.], [#math.equation(block: false, alt: "−2 x plus 2")[$−2 x + 2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "9 minus 3 open parenthesis x plus 2 close parenthesis .")[$9 − 3 ( x + 2 ) .$]

#solutionbox[
#math.equation(block: true, alt: "3 minus 3 x")[$3 − 3 x$]
]
]

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

#solutionbox[
#math.equation(block: true, alt: "2 x minus 20")[$2 x − 20$]
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "4 open parenthesis x minus 8 close parenthesis minus open parenthesis x plus 3 close parenthesis .")[$4 ( x − 8 ) − ( x + 3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 open parenthesis x minus 8 close parenthesis minus open parenthesis x plus 3 close parenthesis")[$4 ( x − 8 ) − ( x + 3 )$]]),
  [Distribute.], [#math.equation(block: false, alt: "4 x minus 32 minus x minus 3")[$4 x − 32 − x − 3$]],
  [Combine like terms.], [#math.equation(block: false, alt: "3 x minus 35")[$3 x − 35$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "6 open parenthesis x minus 9 close parenthesis minus open parenthesis x plus 12 close parenthesis .")[$6 ( x − 9 ) − ( x + 12 ) .$]

#solutionbox[
#math.equation(block: true, alt: "5 x minus 66")[$5 x − 66$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "8 open parenthesis x minus 1 close parenthesis minus open parenthesis x plus 5 close parenthesis .")[$8 ( x − 1 ) − ( x + 5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "7 x minus 13")[$7 x − 13$]
]
]

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

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#strong[Commutative Property] #linebreak() When adding or multiplying, changing the #emph[order] gives the same result #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition If a , b are real numbers, then, a plus b, equals, b plus a; of multiplication If a , b are real numbers, then, a times b, equals, b times a")[$#h(2em) "of addition" #h(0.2em) "If" #h(0.2em) a , b #h(0.2em) "are real numbers, then" &  &  & #h(9.1em) a + b & = & b + a \ #h(2em) "of multiplication" #h(0.2em) "If" #h(0.2em) a , b #h(0.2em) "are real numbers, then" &  &  & #h(9.1em) a · b & = & b · a$]],
  [#strong[Associative Property] #linebreak() When adding or multiplying, changing the #emph[grouping] gives the same result. #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition If a , b , and c are real numbers, then, open parenthesis a plus b close parenthesis plus c, equals, a plus open parenthesis b plus c close parenthesis; of multiplication If a , b , and c are real numbers, then, open parenthesis a times b close parenthesis times c, equals, a times open parenthesis b times c close parenthesis")[$#h(2em) "of addition" #h(0.2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(4em) ( a + b ) + c & = & a + ( b + c ) \ #h(2em) "of multiplication" #h(0.2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(4em) ( a · b ) · c & = & a · ( b · c )$]],
  [#strong[Distributive Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "If a , b , and c are real numbers, then, a open parenthesis b plus c close parenthesis, equals, a b plus a c; open parenthesis b plus c close parenthesis a, equals, b a plus c a; a open parenthesis b minus c close parenthesis, equals, a b minus a c; open parenthesis b minus c close parenthesis a, equals, b a minus c a")[$#h(2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(12.4em) a ( b + c ) & = & a b + a c \  \  &  &  & #h(12.4em) ( b + c ) a & = & b a + c a \  \  &  &  & #h(12.4em) a ( b − c ) & = & a b − a c \  \  &  &  & #h(12.4em) ( b − c ) a & = & b a − c a$]],
  [#strong[Identity Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition For any real number a :, a plus 0 equals a; 0 is the additive identity, 0 plus a equals a; of multiplication For any real number a :, a times 1 equals a; 1 is the multiplicative identity, 1 times a equals a")[$#h(2em) "of addition" #h(0.2em) "For any real number" #h(0.2em) a ":" &  &  & #h(12.1em) a + 0 = a \ #h(4em) 0 #h(0.2em) "is the" #h(0.2em) "additive identity" &  &  & #h(12.1em) 0 + a = a \ #h(2em) "of multiplication" #h(0.2em) "For any real number" #h(0.2em) a ":" &  &  & #h(12.65em) a · 1 = a \ #h(4em) 1 #h(0.2em) "is the" #h(0.2em) "multiplicative identity" &  &  & #h(12.65em) 1 · a = a$]],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#strong[Inverse Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition For any real number a ,, a plus open parenthesis − a close parenthesis equals 0; − a is the additive inverse of a; A number and its o p p o s i t e add to zero.; of multiplication For any real number a , a not equal to 0, a times the fraction 1 over a equals 1; the fraction 1 over a is the multiplicative inverse of a; A number and its r e c i p r o c a l multiply to one.")[$#h(2em) "of addition" #h(0.2em) "For any real number" #h(0.2em) a , &  &  & #h(4.4em) a + ( "−" a ) = 0 \ #h(4em) "−" a #h(0.2em) "is the" #h(0.2em) "additive inverse" #h(0.2em) "of" #h(0.2em) a &  &  &  \ #h(4em) "A number and its" #h(0.2em) o p p o s i t e #h(0.2em) "add to zero." &  &  &  \ #h(2em) "of multiplication" #h(0.2em) "For any real number" #h(0.2em) a , a ≠ 0 &  &  & #h(4.7em) a · frac(1, a) = 1 \  \ #h(4em) frac(1, a) #h(0.2em) "is the" #h(0.2em) "multiplicative inverse" #h(0.2em) "of" #h(0.2em) a &  &  &  \ #h(4em) "A number and its" #h(0.2em) r e c i p r o c a l #h(0.2em) "multiply to one." &  &  &$]],
  [#strong[Properties of Zero] #linebreak() 
#math.equation(block: false, alt: "For any real number a ,, a times 0 equals 0; 0 times a equals 0; For any real number a , a not equal to 0 ,, the fraction 0 over a equals 0; For any real number a ,, the fraction a over 0 is undefined")[$#h(2em) "For any real number" #h(0.2em) a , &  &  & #h(14em) a · 0 = 0 \  &  &  & #h(14em) 0 · a = 0 \ #h(2em) "For any real number" #h(0.2em) a , a ≠ 0 , &  &  & #h(15em) frac(0, a) = 0 \ #h(2em) "For any real number" #h(0.2em) a , &  &  & #h(13em) frac(a, 0) #h(0.2em) "is undefined"$]],
))


=== Key Concepts

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#strong[Commutative Property] #linebreak() When adding or multiplying, changing the #emph[order] gives the same result #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition If a , b are real numbers, then, a plus b, equals, b plus a; of multiplication If a , b are real numbers, then, a times b, equals, b times a")[$#h(2em) "of addition" #h(0.2em) "If" #h(0.2em) a , b #h(0.2em) "are real numbers, then" &  &  & #h(9.1em) a + b & = & b + a \ #h(2em) "of multiplication" #h(0.2em) "If" #h(0.2em) a , b #h(0.2em) "are real numbers, then" &  &  & #h(9.1em) a · b & = & b · a$]],
  [#strong[Associative Property] #linebreak() When adding or multiplying, changing the #emph[grouping] gives the same result. #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition If a , b , and c are real numbers, then, open parenthesis a plus b close parenthesis plus c, equals, a plus open parenthesis b plus c close parenthesis; of multiplication If a , b , and c are real numbers, then, open parenthesis a times b close parenthesis times c, equals, a times open parenthesis b times c close parenthesis")[$#h(2em) "of addition" #h(0.2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(4em) ( a + b ) + c & = & a + ( b + c ) \ #h(2em) "of multiplication" #h(0.2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(4em) ( a · b ) · c & = & a · ( b · c )$]],
  [#strong[Distributive Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "If a , b , and c are real numbers, then, a open parenthesis b plus c close parenthesis, equals, a b plus a c; open parenthesis b plus c close parenthesis a, equals, b a plus c a; a open parenthesis b minus c close parenthesis, equals, a b minus a c; open parenthesis b minus c close parenthesis a, equals, b a minus c a")[$#h(2em) "If" #h(0.2em) a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers, then" &  &  & #h(12.4em) a ( b + c ) & = & a b + a c \  \  &  &  & #h(12.4em) ( b + c ) a & = & b a + c a \  \  &  &  & #h(12.4em) a ( b − c ) & = & a b − a c \  \  &  &  & #h(12.4em) ( b − c ) a & = & b a − c a$]],
  [#strong[Identity Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition For any real number a :, a plus 0 equals a; 0 is the additive identity, 0 plus a equals a; of multiplication For any real number a :, a times 1 equals a; 1 is the multiplicative identity, 1 times a equals a")[$#h(2em) "of addition" #h(0.2em) "For any real number" #h(0.2em) a ":" &  &  & #h(12.1em) a + 0 = a \ #h(4em) 0 #h(0.2em) "is the" #h(0.2em) "additive identity" &  &  & #h(12.1em) 0 + a = a \ #h(2em) "of multiplication" #h(0.2em) "For any real number" #h(0.2em) a ":" &  &  & #h(12.65em) a · 1 = a \ #h(4em) 1 #h(0.2em) "is the" #h(0.2em) "multiplicative identity" &  &  & #h(12.65em) 1 · a = a$]],
  [#strong[Inverse Property] #linebreak()  #linebreak() 
#math.equation(block: false, alt: "of addition For any real number a ,, a plus open parenthesis − a close parenthesis equals 0; − a is the additive inverse of a; A number and its o p p o s i t e add to zero.; of multiplication For any real number a , a not equal to 0, a times the fraction 1 over a equals 1; the fraction 1 over a is the multiplicative inverse of a; A number and its r e c i p r o c a l multiply to one.")[$#h(2em) "of addition" #h(0.2em) "For any real number" #h(0.2em) a , &  &  & #h(7.1em) a + ( "−" a ) = 0 \ #h(4em) "−" a #h(0.2em) "is the" #h(0.2em) "additive inverse" #h(0.2em) "of" #h(0.2em) a &  &  &  \ #h(4em) "A number and its" #h(0.2em) o p p o s i t e #h(0.2em) "add to zero." &  &  &  \ #h(2em) "of multiplication" #h(0.2em) "For any real number" #h(0.2em) a , a ≠ 0 &  &  & #h(7.4em) a · frac(1, a) = 1 \  \ #h(4em) frac(1, a) #h(0.2em) "is the" #h(0.2em) "multiplicative inverse" #h(0.2em) "of" #h(0.2em) a &  &  &  \ #h(4em) "A number and its" #h(0.2em) r e c i p r o c a l #h(0.2em) "multiply to one." &  &  &$]],
  [#strong[Properties of Zero] #linebreak() 
#math.equation(block: false, alt: "For any real number a ,, a times 0 equals 0; 0 times a equals 0; For any real number a , a not equal to 0 ,, the fraction 0 over a equals 0; For any real number a ,, the fraction a over 0 is undefined")[$#h(2em) "For any real number" #h(0.2em) a , &  &  & #h(16.7em) a · 0 = 0 \  &  &  & #h(16.7em) 0 · a = 0 \ #h(2em) "For any real number" #h(0.2em) a , a ≠ 0 , &  &  & #h(17.7em) frac(0, a) = 0 \ #h(2em) "For any real number" #h(0.2em) a , &  &  & #h(15.7em) frac(a, 0) #h(0.2em) "is undefined"$]],
))


=== Section Exercises


==== Practice Makes Perfect

#strong[Use the Commutative and Associative Properties]

In the following exercises, simplify.

#math.equation(block: true, alt: "43 m plus open parenthesis −12 n close parenthesis plus open parenthesis −16 m close parenthesis plus open parenthesis −9 n close parenthesis")[$43 m + ( −12 n ) + ( −16 m ) + ( −9 n )$]

#solutionbox[
#math.equation(block: true, alt: "27 m plus open parenthesis −21 n close parenthesis")[$27 m + ( −21 n )$]
]

#math.equation(block: true, alt: "−22 p plus 17 q plus open parenthesis −35 p close parenthesis plus open parenthesis −27 q close parenthesis")[$−22 p + 17 q + ( −35 p ) + ( −27 q )$]

#math.equation(block: true, alt: "the fraction 3 over 8 g plus the fraction 1 over 12 h plus the fraction 7 over 8 g plus the fraction 5 over 12 h")[$frac(3, 8) g + frac(1, 12) h + frac(7, 8) g + frac(5, 12) h$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 4 g plus the fraction 1 over 2 h")[$frac(5, 4) g + frac(1, 2) h$]
]

#math.equation(block: true, alt: "the fraction 5 over 6 a plus the fraction 3 over 10 b plus the fraction 1 over 6 a plus the fraction 9 over 10 b")[$frac(5, 6) a + frac(3, 10) b + frac(1, 6) a + frac(9, 10) b$]

#math.equation(block: true, alt: "6.8 p plus 9.14 q plus open parenthesis −4.37 p close parenthesis plus open parenthesis −0.88 q close parenthesis")[$6.8 p + 9.14 q + ( −4.37 p ) + ( −0.88 q )$]

#solutionbox[
#math.equation(block: true, alt: "2.43 p plus 8.26 q")[$2.43 p + 8.26 q$]
]

#math.equation(block: true, alt: "9.6 m plus 7.22 n plus open parenthesis −2.19 m close parenthesis plus open parenthesis −0.65 n close parenthesis")[$9.6 m + 7.22 n + ( −2.19 m ) + ( −0.65 n )$]

#math.equation(block: true, alt: "−24 times 7 times the fraction 3 over 8")[$−24 · 7 · frac(3, 8)$]

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

#math.equation(block: true, alt: "−36 times 11 times the fraction 4 over 9")[$−36 · 11 · frac(4, 9)$]

#math.equation(block: true, alt: "open parenthesis the fraction 5 over 6 plus the fraction 8 over 15 close parenthesis plus the fraction 7 over 15")[$( frac(5, 6) + frac(8, 15) ) + frac(7, 15)$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 5 over 6")[$1 frac(5, 6)$]
]

#math.equation(block: true, alt: "open parenthesis the fraction 11 over 12 plus the fraction 4 over 9 close parenthesis plus the fraction 5 over 9")[$( frac(11, 12) + frac(4, 9) ) + frac(5, 9)$]

#math.equation(block: true, alt: "17 open parenthesis 0.25 close parenthesis open parenthesis 4 close parenthesis")[$17 ( 0.25 ) ( 4 )$]

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

#math.equation(block: true, alt: "36 open parenthesis 0.2 close parenthesis open parenthesis 5 close parenthesis")[$36 ( 0.2 ) ( 5 )$]

#math.equation(block: true, alt: "[ 2.48 open parenthesis 12 close parenthesis ] open parenthesis 0.5 close parenthesis")[$[ 2.48 ( 12 ) ] ( 0.5 )$]

#solutionbox[
#math.equation(block: true, alt: "14.88")[$14.88$]
]

#math.equation(block: true, alt: "[ 9.731 open parenthesis 4 close parenthesis ] open parenthesis 0.75 close parenthesis")[$[ 9.731 ( 4 ) ] ( 0.75 )$]

#math.equation(block: true, alt: "12 open parenthesis the fraction 5 over 6 p close parenthesis")[$12 ( frac(5, 6) p )$]

#solutionbox[
#math.equation(block: true, alt: "10 p")[$10 p$]
]

#math.equation(block: true, alt: "20 open parenthesis the fraction 3 over 5 q close parenthesis")[$20 ( frac(3, 5) q )$]

#strong[Use the Properties of Identity, Inverse and Zero]

In the following exercises, simplify.

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

#solutionbox[
#math.equation(block: true, alt: "44")[$44$]
]

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

#math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 7 over 8 plus open parenthesis minus the fraction 1 over 2 close parenthesis")[$frac(1, 2) + frac(7, 8) + ( − frac(1, 2) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 8")[$frac(7, 8)$]
]

#math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 5 over 12 plus open parenthesis minus the fraction 2 over 5 close parenthesis")[$frac(2, 5) + frac(5, 12) + ( − frac(2, 5) )$]

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

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

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

#math.equation(block: true, alt: "the fraction 3 over 20 times the fraction 49 over 11 times the fraction 20 over 3")[$frac(3, 20) · frac(49, 11) · frac(20, 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 49 over 11")[$frac(49, 11)$]
]

#math.equation(block: true, alt: "the fraction 13 over 18 times the fraction 25 over 7 times the fraction 18 over 13")[$frac(13, 18) · frac(25, 7) · frac(18, 13)$]

#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")[$u ≠ 4.99$]

#solutionbox[
0
]

#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: "x not equal to the fraction 1 over 6")[$x ≠ frac(1, 6)$]

#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")[$32 − 5 a ≠ 0$]

#solutionbox[
undefined
]

#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")[$28 − 9 b ≠ 0$]

#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")[$frac(3, 4) + frac(9, 10) m ≠ 0$]

#solutionbox[
undefined
]

#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")[$frac(5, 16) n − frac(3, 7) ≠ 0$]

#strong[Simplify Expressions Using the Distributive Property]

In the following exercises, simplify using the Distributive Property.

#math.equation(block: true, alt: "8 open parenthesis 4 y plus 9 close parenthesis")[$8 ( 4 y + 9 )$]

#solutionbox[
#math.equation(block: true, alt: "32 y plus 72")[$32 y + 72$]
]

#math.equation(block: true, alt: "9 open parenthesis 3 w plus 7 close parenthesis")[$9 ( 3 w + 7 )$]

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

#solutionbox[
#math.equation(block: true, alt: "6 c minus 78")[$6 c − 78$]
]

#math.equation(block: true, alt: "7 open parenthesis y minus 13 close parenthesis")[$7 ( y − 13 )$]

#math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis 3 q plus 12 close parenthesis")[$frac(1, 4) ( 3 q + 12 )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 4 q plus 3")[$frac(3, 4) q + 3$]
]

#math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis 4 m plus 20 close parenthesis")[$frac(1, 5) ( 4 m + 20 )$]

#math.equation(block: true, alt: "9 open parenthesis the fraction 5 over 9 y minus the fraction 1 over 3 close parenthesis")[$9 ( frac(5, 9) y − frac(1, 3) )$]

#solutionbox[
#math.equation(block: true, alt: "5 y minus 3")[$5 y − 3$]
]

#math.equation(block: true, alt: "10 open parenthesis the fraction 3 over 10 x minus the fraction 2 over 5 close parenthesis")[$10 ( frac(3, 10) x − frac(2, 5) )$]

#math.equation(block: true, alt: "12 open parenthesis the fraction 1 over 4 plus the fraction 2 over 3 r close parenthesis")[$12 ( frac(1, 4) + frac(2, 3) r )$]

#solutionbox[
#math.equation(block: true, alt: "3 plus 8 r")[$3 + 8 r$]
]

#math.equation(block: true, alt: "12 open parenthesis the fraction 1 over 6 plus the fraction 3 over 4 s close parenthesis")[$12 ( frac(1, 6) + frac(3, 4) s )$]

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

#solutionbox[
#math.equation(block: true, alt: "36 d plus 90")[$36 d + 90$]
]

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

#math.equation(block: true, alt: "r open parenthesis s minus 18 close parenthesis")[$r ( s − 18 )$]

#solutionbox[
#math.equation(block: true, alt: "r s minus 18 r")[$r s − 18 r$]
]

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

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

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

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

#math.equation(block: true, alt: "−7 open parenthesis 4 p plus 1 close parenthesis")[$−7 ( 4 p + 1 )$]

#solutionbox[
#math.equation(block: true, alt: "−28 p minus 7")[$−28 p − 7$]
]

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

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

#solutionbox[
#math.equation(block: true, alt: "−3 x plus 18")[$−3 x + 18$]
]

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

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

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

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

#math.equation(block: true, alt: "16 minus 3 open parenthesis y plus 8 close parenthesis")[$16 − 3 ( y + 8 )$]

#solutionbox[
#math.equation(block: true, alt: "−3 y minus 8")[$−3 y − 8$]
]

#math.equation(block: true, alt: "18 minus 4 open parenthesis x plus 2 close parenthesis")[$18 − 4 ( x + 2 )$]

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

#solutionbox[
#math.equation(block: true, alt: "−33 c plus 26")[$−33 c + 26$]
]

#math.equation(block: true, alt: "9 minus 6 open parenthesis 7 n minus 5 close parenthesis")[$9 − 6 ( 7 n − 5 )$]

#math.equation(block: true, alt: "22 minus open parenthesis a plus 3 close parenthesis")[$22 − ( a + 3 )$]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "4 m minus 10")[$4 m − 10$]
]

#math.equation(block: true, alt: "open parenthesis 4 y minus 1 close parenthesis minus open parenthesis y minus 2 close parenthesis")[$( 4 y − 1 ) − ( y − 2 )$]

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

#solutionbox[
#math.equation(block: true, alt: "72 x minus 25")[$72 x − 25$]
]

#math.equation(block: true, alt: "4 open parenthesis 6 x minus 1 close parenthesis minus open parenthesis −8 close parenthesis")[$4 ( 6 x − 1 ) − ( −8 )$]

#math.equation(block: true, alt: "5 open parenthesis 2 n plus 9 close parenthesis plus 12 open parenthesis n minus 3 close parenthesis")[$5 ( 2 n + 9 ) + 12 ( n − 3 )$]

#solutionbox[
#math.equation(block: true, alt: "22 n plus 9")[$22 n + 9$]
]

#math.equation(block: true, alt: "9 open parenthesis 5 u plus 8 close parenthesis plus 2 open parenthesis u minus 6 close parenthesis")[$9 ( 5 u + 8 ) + 2 ( u − 6 )$]

#math.equation(block: true, alt: "14 open parenthesis c minus 1 close parenthesis minus 8 open parenthesis c minus 6 close parenthesis")[$14 ( c − 1 ) − 8 ( c − 6 )$]

#solutionbox[
#math.equation(block: true, alt: "6 c plus 34")[$6 c + 34$]
]

#math.equation(block: true, alt: "11 open parenthesis n minus 7 close parenthesis minus 5 open parenthesis n minus 1 close parenthesis")[$11 ( n − 7 ) − 5 ( n − 1 )$]

#math.equation(block: true, alt: "6 open parenthesis 7 y plus 8 close parenthesis minus open parenthesis 30 y minus 15 close parenthesis")[$6 ( 7 y + 8 ) − ( 30 y − 15 )$]

#solutionbox[
#math.equation(block: true, alt: "12 y plus 63")[$12 y + 63$]
]

#math.equation(block: true, alt: "7 open parenthesis 3 n plus 9 close parenthesis minus open parenthesis 4 n minus 13 close parenthesis")[$7 ( 3 n + 9 ) − ( 4 n − 13 )$]


==== Writing Exercises

In your own words, state the Associative Property of addition.

#solutionbox[
Answers will vary.
]

What is the difference between the additive inverse and the multiplicative inverse of a number?

Simplify #math.equation(block: false, alt: "8 open parenthesis x minus the fraction 1 over 4 close parenthesis")[$8 ( x − frac(1, 4) )$] using the Distributive Property and explain each step.

#solutionbox[
Answers will vary.
]

Explain how you can multiply #math.equation(block: false, alt: "4 open parenthesis $ 5.97 close parenthesis")[$4 ( "$" 5.97 )$] without paper or calculator by thinking of #math.equation(block: false, alt: "$ 5.97")[$"$" 5.97$] as #math.equation(block: false, alt: "6 minus 0.03")[$6 − 0.03$] and then using the Distributive Property.


==== Self Check

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

#figure(figph[This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: use the commutative and associative properties, use the properties of identity, inverse and zero, simplify expressions using the Distributive Property. The remaining columns are blank.], alt: "This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: use the commutative and associative properties, use the properties of identity, inverse and zero, simplify expressions using the Distributive Property. The remaining columns are blank.", caption: none)

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


=== Chapter Review Exercises


==== Use the Language of Algebra

#strong[Identify Multiples and Factors]

Use the divisibility tests to determine whether 180 is divisible by 2, by 3, by 5, by 6, and by 10.

#solutionbox[
Divisible by #math.equation(block: false, alt: "2 , 3 , 5 , 6 , 10")[$2 , 3 , 5 , 6 , 10$]
]

Find the prime factorization of 252.

Find the least common multiple of 24 and 40.

#solutionbox[
120
]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "24 divided by 3 plus 4 open parenthesis 5 minus 2 close parenthesis")[$24 ÷ 3 + 4 ( 5 − 2 )$]

#math.equation(block: true, alt: "7 plus 3 [ 6 minus 4 open parenthesis 5 minus 4 close parenthesis ] minus 3 squared")[$7 + 3 [ 6 − 4 ( 5 − 4 ) ] − 3^(2)$]

#solutionbox[
4
]

#strong[Evaluate an Expression]

In the following exercises, evaluate the following expressions.

When #math.equation(block: false, alt: "x equals 4 ,")[$x = 4 ,$] ⓐ #math.equation(block: false, alt: "x cubed")[$x^(3)$] ⓑ #math.equation(block: false, alt: "5 to the power x")[$5^(x)$] ⓒ #math.equation(block: false, alt: "2 x squared minus 5 x plus 3")[$2 x^(2) − 5 x + 3$]

#math.equation(block: false, alt: "2 x squared minus 4 x y minus 3 y squared")[$2 x^(2) − 4 x y − 3 y^(2)$] when #math.equation(block: false, alt: "x equals 3 ,")[$x = 3 ,$] #math.equation(block: false, alt: "y equals 1")[$y = 1$]

#solutionbox[
3
]

#strong[Simplify Expressions by Combining Like Terms]

In the following exercises, simplify the following expressions by combining like terms.

#math.equation(block: true, alt: "12 y plus 7 plus 2 y minus 5")[$12 y + 7 + 2 y − 5$]

#math.equation(block: true, alt: "14 x squared minus 9 x plus 11 minus 8 x squared plus 8 x minus 6")[$14 x^(2) − 9 x + 11 − 8 x^(2) + 8 x − 6$]

#solutionbox[
#math.equation(block: true, alt: "6 x squared minus x plus 5")[$6 x^(2) − x + 5$]
]

#strong[Translate an English Phrase to an Algebraic Expression]

In the following exercises, translate the phrases into algebraic expressions.

ⓐ the sum of #math.equation(block: false, alt: "4 a b squared")[$4 a b^(2)$] and #math.equation(block: false, alt: "7 a cubed b squared")[$7 a^(3) b^(2)$] #linebreak() ⓑ the product of #math.equation(block: false, alt: "6 y squared")[$6 y^(2)$] and #math.equation(block: false, alt: "3 y")[$3 y$] #linebreak() ⓒ twelve more than #math.equation(block: false, alt: "5 x")[$5 x$] #linebreak() ⓓ #math.equation(block: false, alt: "5 y")[$5 y$] less than #math.equation(block: false, alt: "8 y squared")[$8 y^(2)$]

ⓐ eleven times the difference of #math.equation(block: false, alt: "y")[$y$] and two #linebreak() ⓑ the difference of eleven times #math.equation(block: false, alt: "y")[$y$] and two

#solutionbox[
ⓐ #math.equation(block: false, alt: "11 open parenthesis y minus 2 close parenthesis")[$11 ( y − 2 )$] ⓑ #math.equation(block: false, alt: "11 y minus 2")[$11 y − 2$]
]

Dushko has nickels and pennies in his pocket. The number of pennies is four less than five times the number of nickels. Let #math.equation(block: false, alt: "n")[$n$] represent the number of nickels. Write an expression for the number of pennies.


==== Integers

#strong[Simplify Expressions with Absolute Value]

In the following exercise, fill in #math.equation(block: false, alt: "less than , greater than ,")[$< , > ,$] or #math.equation(block: false, alt: "equals")[$=$] for each of the following pairs of numbers.

ⓐ #math.equation(block: false, alt: "− | 7 | ___ minus | −7 |")[$"−" | 7 | \_\_\_ − | −7 |$] #linebreak() ⓑ #math.equation(block: false, alt: "−8 ___ minus | −8 |")[$−8 \_\_\_ − | −8 |$] #linebreak() ⓒ #math.equation(block: false, alt: "| −13 | ___ minus 13")[$| −13 | \_\_\_ − 13$] #linebreak() ⓓ #math.equation(block: false, alt: "| −12 | ___ minus open parenthesis −12 close parenthesis")[$| −12 | \_\_\_ − ( −12 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "equals")[$=$] ⓑ #math.equation(block: false, alt: "equals")[$=$] ⓒ #math.equation(block: false, alt: "greater than")[$>$] ⓓ #math.equation(block: false, alt: "equals")[$=$]
]

In the following exercises, simplify.

#math.equation(block: true, alt: "9 minus | 3 open parenthesis 4 minus 8 close parenthesis |")[$9 − | 3 ( 4 − 8 ) |$]

#math.equation(block: true, alt: "12 minus 3 | 1 minus 4 open parenthesis 4 minus 2 close parenthesis |")[$12 − 3 | 1 − 4 ( 4 − 2 ) |$]

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

#strong[Add and Subtract Integers]

In the following exercises, simplify each expression.

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

ⓐ #math.equation(block: false, alt: "15 minus 7")[$15 − 7$] #linebreak() ⓑ #math.equation(block: false, alt: "−15 minus open parenthesis −7 close parenthesis")[$−15 − ( −7 )$] #linebreak() ⓒ #math.equation(block: false, alt: "−15 minus 7")[$−15 − 7$] #linebreak() ⓓ #math.equation(block: false, alt: "15 minus open parenthesis −7 close parenthesis")[$15 − ( −7 )$]

#solutionbox[
ⓐ 8 ⓑ #math.equation(block: false, alt: "−8")[$−8$] ⓒ #math.equation(block: false, alt: "−22")[$−22$] ⓓ 22
]

#math.equation(block: true, alt: "−11 minus open parenthesis −12 close parenthesis plus 5")[$−11 − ( −12 ) + 5$]

ⓐ #math.equation(block: false, alt: "23 minus open parenthesis −17 close parenthesis")[$23 − ( −17 )$] ⓑ #math.equation(block: false, alt: "23 plus 17")[$23 + 17$]

#solutionbox[
ⓐ 40 ⓑ 40
]

#math.equation(block: true, alt: "− open parenthesis 7 minus 11 close parenthesis minus open parenthesis 3 minus 5 close parenthesis")[$"−" ( 7 − 11 ) − ( 3 − 5 )$]

#strong[Multiply and Divide Integers]

In the following exercise, multiply or divide.

ⓐ #math.equation(block: false, alt: "−27 divided by 9")[$−27 ÷ 9$] ⓑ #math.equation(block: false, alt: "120 divided by open parenthesis −8 close parenthesis")[$120 ÷ ( −8 )$] ⓒ #math.equation(block: false, alt: "4 open parenthesis −14 close parenthesis")[$4 ( −14 )$] ⓓ #math.equation(block: false, alt: "−1 open parenthesis −17 close parenthesis")[$−1 ( −17 )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−3")[$−3$] ⓑ #math.equation(block: false, alt: "−15")[$−15$] ⓒ #math.equation(block: false, alt: "−56")[$−56$] ⓓ 17
]

#strong[Simplify and Evaluate Expressions with Integers]

In the following exercises, simplify each expression.

ⓐ #math.equation(block: false, alt: "open parenthesis −7 close parenthesis cubed")[$attach(( −7 ), t: 3)$] ⓑ #math.equation(block: false, alt: "− 7 cubed")[$"−" 7^(3)$]

#math.equation(block: true, alt: "open parenthesis 7 minus 11 close parenthesis open parenthesis 6 minus 13 close parenthesis")[$( 7 − 11 ) ( 6 − 13 )$]

#solutionbox[
28
]

#math.equation(block: true, alt: "63 divided by open parenthesis −9 close parenthesis plus open parenthesis −36 close parenthesis divided by open parenthesis −4 close parenthesis")[$63 ÷ ( −9 ) + ( −36 ) ÷ ( −4 )$]

#math.equation(block: true, alt: "6 minus 3 | 4 open parenthesis 1 minus 2 close parenthesis minus open parenthesis 7 minus 5 close parenthesis |")[$6 − 3 | 4 ( 1 − 2 ) − ( 7 − 5 ) |$]

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

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis to the power 4 minus 24 divided by open parenthesis 13 minus 5 close parenthesis")[$attach(( −2 ), t: 4) − 24 ÷ ( 13 − 5 )$]

For the following exercises, evaluate each expression.

#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 −4 , z equals 7")[$y = −4 , z = 7$]

#solutionbox[
9
]

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

#strong[Translate English Phrases to Algebraic Expressions]

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

the sum of #math.equation(block: false, alt: "−4")[$−4$] and #math.equation(block: false, alt: "−9 ,")[$−9 ,$] increased by 23

#solutionbox[
#math.equation(block: true, alt: "open parenthesis −4 plus open parenthesis −9 close parenthesis close parenthesis plus 23 ; 10")[$( −4 + ( −9 ) ) + 23 ; 10$]
]

ⓐ the difference of 17 and #math.equation(block: false, alt: "−8")[$−8$] ⓑ subtract 17 from #math.equation(block: false, alt: "−25")[$−25$]

#strong[Use Integers in Applications]

In the following exercise, solve.

#strong[Temperature] On July 10, the high temperature in Phoenix, Arizona, was 109°, and the high temperature in Juneau, Alaska, was 63°. What was the difference between the temperature in Phoenix and the temperature in Juneau?

#solutionbox[
#math.equation(block: true, alt: "46 degrees")[$46^("°")$]
]


==== Fractions

#strong[Simplify Fractions]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 204 over 228")[$frac(204, 228)$]

#math.equation(block: true, alt: "minus the fraction 270 x cubed over 198 y squared")[$− frac(270 x^(3), 198 y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 15 x cubed over 11 y squared")[$− frac(15 x^(3), 11 y^(2))$]
]

#strong[Multiply and Divide Fractions]

In the following exercises, perform the indicated operation.

#math.equation(block: true, alt: "open parenthesis minus the fraction 14 over 15 close parenthesis open parenthesis the fraction 10 over 21 close parenthesis")[$( − frac(14, 15) ) ( frac(10, 21) )$]

#math.equation(block: true, alt: "the fraction 6 x over 25 divided by the fraction 9 y over 20")[$frac(6 x, 25) ÷ frac(9 y, 20)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 x over 15 y")[$frac(8 x, 15 y)$]
]

#math.equation(block: true, alt: "the fraction minus the fraction 4 over 9 over the fraction 8 over 21")[$frac(− frac(4, 9), frac(8, 21))$]

#strong[Add and Subtract Fractions]

In the following exercises, perform the indicated operation.

#math.equation(block: true, alt: "the fraction 5 over 18 plus the fraction 7 over 12")[$frac(5, 18) + frac(7, 12)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 31 over 36")[$frac(31, 36)$]
]

#math.equation(block: true, alt: "the fraction 11 over 36 minus the fraction 15 over 48")[$frac(11, 36) − frac(15, 48)$]

ⓐ #math.equation(block: false, alt: "the fraction 5 over 8 plus the fraction 3 over 4")[$frac(5, 8) + frac(3, 4)$] ⓑ #math.equation(block: false, alt: "the fraction 5 over 8 divided by the fraction 3 over 4")[$frac(5, 8) ÷ frac(3, 4)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 11 over 8")[$frac(11, 8)$] ⓑ #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$]
]

ⓐ #math.equation(block: false, alt: "minus the fraction 3 y over 10 minus the fraction 5 over 6")[$− frac(3 y, 10) − frac(5, 6)$] ⓑ #math.equation(block: false, alt: "minus the fraction 3 y over 10 times the fraction 5 over 6")[$− frac(3 y, 10) · frac(5, 6)$]

#strong[Use the Order of Operations to Simplify Fractions]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 4 times 3 minus 2 times 5 over −6 times 3 plus 2 times 3")[$frac(4 · 3 − 2 · 5, −6 · 3 + 2 · 3)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 6")[$− frac(1, 6)$]
]

#math.equation(block: true, alt: "the fraction 4 open parenthesis 7 minus 3 close parenthesis minus 2 open parenthesis 4 minus 9 close parenthesis over −3 open parenthesis 4 plus 2 close parenthesis plus 7 open parenthesis 3 minus 6 close parenthesis")[$frac(4 ( 7 − 3 ) − 2 ( 4 − 9 ), −3 ( 4 + 2 ) + 7 ( 3 − 6 ))$]

#math.equation(block: true, alt: "the fraction 4 cubed minus 4 squared over open parenthesis the fraction 4 over 5 close parenthesis squared")[$frac(4^(3) − 4^(2), attach(( frac(4, 5) ), t: 2))$]

#solutionbox[
75
]

#strong[Evaluate Variable Expressions with Fractions]

In the following exercises, evaluate.

#math.equation(block: false, alt: "4 x squared y squared")[$4 x^(2) y^(2)$] when #linebreak() #math.equation(block: false, alt: "x equals the fraction 2 over 3")[$x = frac(2, 3)$] and #math.equation(block: false, alt: "y equals minus the fraction 3 over 4")[$y = − frac(3, 4)$]

#math.equation(block: false, alt: "the fraction a plus b over a minus b")[$frac(a + b, a − b)$] when #linebreak() #math.equation(block: false, alt: "a equals −4 ,")[$a = −4 ,$] #math.equation(block: false, alt: "b equals 6")[$b = 6$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 1 over 5")[$− frac(1, 5)$]
]


==== Decimals

#strong[Round Decimals]

Round #math.equation(block: false, alt: "6.738")[$6.738$] to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

#strong[Add and Subtract Decimals]

In the following exercises, perform the indicated operation.

#math.equation(block: true, alt: "−23.67 plus 29.84")[$−23.67 + 29.84$]

#solutionbox[
#math.equation(block: true, alt: "6.17")[$6.17$]
]

#math.equation(block: true, alt: "54.3 minus 100")[$54.3 − 100$]

#math.equation(block: true, alt: "79.38 minus open parenthesis −17.598 close parenthesis")[$79.38 − ( −17.598 )$]

#solutionbox[
#math.equation(block: true, alt: "96.978")[$96.978$]
]

#strong[Multiply and Divide Decimals]

In the following exercises, perform the indicated operation.

#math.equation(block: true, alt: "open parenthesis −2.8 close parenthesis open parenthesis 3.97 close parenthesis")[$( −2.8 ) ( 3.97 )$]

#math.equation(block: true, alt: "open parenthesis −8.43 close parenthesis open parenthesis −57.91 close parenthesis")[$( −8.43 ) ( −57.91 )$]

#solutionbox[
488.1813
]

#math.equation(block: true, alt: "open parenthesis 53.48 close parenthesis open parenthesis 10 close parenthesis")[$( 53.48 ) ( 10 )$]

#math.equation(block: true, alt: "open parenthesis 0.563 close parenthesis open parenthesis 100 close parenthesis")[$( 0.563 ) ( 100 )$]

#solutionbox[
#math.equation(block: true, alt: "56.3")[$56.3$]
]

#math.equation(block: true, alt: "$ 118.35 divided by 2.6")[$"$" 118.35 ÷ 2.6$]

#math.equation(block: true, alt: "1.84 divided by open parenthesis −0.8 close parenthesis")[$1.84 ÷ ( −0.8 )$]

#solutionbox[
#math.equation(block: true, alt: "−2.3")[$−2.3$]
]

#strong[Convert Decimals, Fractions and Percents]

In the following exercises, convert each decimal to a fraction.

#math.equation(block: true, alt: "0.65")[$0.65$]

#math.equation(block: true, alt: "−9.6")[$−9.6$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 48 over 5")[$− frac(48, 5)$]
]

In the following exercises, convert each fraction to a decimal.

#math.equation(block: true, alt: "minus the fraction 5 over 8")[$− frac(5, 8)$]

#math.equation(block: true, alt: "the fraction 14 over 11")[$frac(14, 11)$]

#solutionbox[
#math.equation(block: true, alt: "1. 27 ¯")[$1. overline(27)$]
]

In the following exercises, convert each decimal to a percent.

#math.equation(block: true, alt: "2.43")[$2.43$]

#math.equation(block: true, alt: "0.0475")[$0.0475$]

#solutionbox[
#math.equation(block: true, alt: "4.75 %")[$4.75 %$]
]

#strong[Simplify Expressions with Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 289")[$sqrt(289)$]

#math.equation(block: true, alt: "the square root of −121")[$sqrt(−121)$]

#solutionbox[
no real number
]

#strong[Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers]

In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers

#math.equation(block: true, alt: "−8 , 0 , 1.95286... , the fraction 12 over 5 , the square root of 36 , 9")[$−8 , 0 , 1.95286... , frac(12, 5) , sqrt(36) , 9$]

#strong[Locate Fractions and Decimals on the Number Line]

In the following exercises, locate the numbers on a number line.

#math.equation(block: true, alt: "the fraction 3 over 4 , minus the fraction 3 over 4 , 1 the fraction 1 over 3 , −1 the fraction 2 over 3 , the fraction 7 over 2 , minus the fraction 5 over 2")[$frac(3, 4) , − frac(3, 4) , 1 frac(1, 3) , −1 frac(2, 3) , frac(7, 2) , − frac(5, 2)$]

#solutionbox[
#figure(figph[Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.], alt: "Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.", caption: none)
]

ⓐ #math.equation(block: false, alt: "3.2")[$3.2$] ⓑ #math.equation(block: false, alt: "−1.35")[$−1.35$]


==== Properties of Real Numbers

#strong[Use the Commutative and Associative Properties]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 5 over 8 x plus the fraction 5 over 12 y plus the fraction 1 over 8 x plus the fraction 7 over 12 y")[$frac(5, 8) x + frac(5, 12) y + frac(1, 8) x + frac(7, 12) y$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 4 x plus y")[$frac(3, 4) x + y$]
]

#math.equation(block: true, alt: "−32 times 9 times the fraction 5 over 8")[$−32 · 9 · frac(5, 8)$]

#math.equation(block: true, alt: "open parenthesis the fraction 11 over 15 plus the fraction 3 over 8 close parenthesis plus the fraction 5 over 8")[$( frac(11, 15) + frac(3, 8) ) + frac(5, 8)$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 11 over 15")[$1 frac(11, 15)$]
]

#strong[Use the Properties of Identity, Inverse and Zero]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction 4 over 7 plus the fraction 8 over 15 plus open parenthesis minus the fraction 4 over 7 close parenthesis")[$frac(4, 7) + frac(8, 15) + ( − frac(4, 7) )$]

#math.equation(block: true, alt: "the fraction 13 over 15 times the fraction 9 over 17 times the fraction 15 over 13")[$frac(13, 15) · frac(9, 17) · frac(15, 13)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 9 over 17")[$frac(9, 17)$]
]

#math.equation(block: true, alt: "the fraction 0 over x minus 3 , x not equal to 3")[$frac(0, x − 3) , x ≠ 3$]

#math.equation(block: true, alt: "the fraction 5 x minus 7 over 0 , 5 x minus 7 not equal to 0")[$frac(5 x − 7, 0) , 5 x − 7 ≠ 0$]

#solutionbox[
undefined
]

#strong[Simplify Expressions Using the Distributive Property]

In the following exercises, simplify using the Distributive Property.

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

#math.equation(block: true, alt: "12 open parenthesis the fraction 2 over 3 b plus the fraction 5 over 6 close parenthesis")[$12 ( frac(2, 3) b + frac(5, 6) )$]

#solutionbox[
#math.equation(block: true, alt: "8 b plus 10")[$8 b + 10$]
]

#math.equation(block: true, alt: "18 times the fraction 5 over 6 open parenthesis 2 x minus 5 close parenthesis")[$18 · frac(5, 6) ( 2 x − 5 )$]

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

#solutionbox[
#math.equation(block: true, alt: "x p minus 5 p")[$x p − 5 p$]
]

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

#math.equation(block: true, alt: "12 minus 6 open parenthesis x plus 3 close parenthesis")[$12 − 6 ( x + 3 )$]

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

#math.equation(block: true, alt: "6 open parenthesis 3 x minus 4 close parenthesis minus open parenthesis −5 close parenthesis")[$6 ( 3 x − 4 ) − ( −5 )$]

#math.equation(block: true, alt: "5 open parenthesis 2 y plus 3 close parenthesis minus open parenthesis 4 y minus 1 close parenthesis")[$5 ( 2 y + 3 ) − ( 4 y − 1 )$]

#solutionbox[
#math.equation(block: true, alt: "6 y plus 16")[$6 y + 16$]
]


=== Practice Test

Find the prime factorization of #math.equation(block: false, alt: "756 .")[$756 .$]

Combine like terms: #math.equation(block: false, alt: "5 n plus 8 plus 2 n minus 1")[$5 n + 8 + 2 n − 1$]

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

Evaluate when #math.equation(block: false, alt: "x equals −2")[$x = −2$] and #math.equation(block: false, alt: "y equals 3 :")[$y = 3 :$] #math.equation(block: false, alt: "the fraction | 3 x minus 4 y | over 6")[$frac(| 3 x − 4 y |, 6)$]

Translate to an algebraic expression and simplify:

ⓐ eleven less than negative eight

ⓑ the difference of #math.equation(block: false, alt: "−8")[$−8$] and #math.equation(block: false, alt: "−3")[$−3$], increased by 5

#solutionbox[
#math.equation(block: true, alt: "−8 minus 11 ; minus 19")[$−8 − 11 ; − 19$]

#math.equation(block: true, alt: "open parenthesis −8 minus open parenthesis −3 close parenthesis close parenthesis plus 5 ; 0")[$( −8 − ( −3 ) ) + 5 ; 0$]
]

Dushko has nickels and pennies in his pocket. The number of pennies is seven less than four times the number of nickels. Let #math.equation(block: false, alt: "n")[$n$] represent the number of nickels. Write an expression for the number of pennies.

Round #math.equation(block: false, alt: "28.1458")[$28.1458$] to the nearest

ⓐ hundredth ⓑ thousandth

#solutionbox[
ⓐ #math.equation(block: false, alt: "28.15")[$28.15$] ⓑ #math.equation(block: false, alt: "28.146")[$28.146$]
]

Convert

ⓐ #math.equation(block: false, alt: "the fraction 5 over 11")[$frac(5, 11)$] to a decimal ⓑ #math.equation(block: false, alt: "1.15")[$1.15$] to a percent

Locate #math.equation(block: false, alt: "the fraction 3 over 5 , 2.8 , and minus the fraction 5 over 2")[$frac(3, 5) , 2.8 , "and" − frac(5, 2)$] on a number line.

#solutionbox[
#figure(figph[Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.], alt: "Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.", caption: none)
]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "8 plus 3 [ 6 minus 3 open parenthesis 5 minus 2 close parenthesis ] minus 4 squared")[$8 + 3 [ 6 − 3 ( 5 − 2 ) ] − 4^(2)$]

#math.equation(block: true, alt: "− open parenthesis 4 minus 9 close parenthesis minus open parenthesis 9 minus 5 close parenthesis")[$"−" ( 4 − 9 ) − ( 9 − 5 )$]

#solutionbox[
1
]

#math.equation(block: true, alt: "56 divided by open parenthesis −8 close parenthesis plus open parenthesis −27 close parenthesis divided by open parenthesis −3 close parenthesis")[$56 ÷ ( −8 ) + ( −27 ) ÷ ( −3 )$]

#math.equation(block: true, alt: "16 minus 2 | 3 open parenthesis 1 minus 4 close parenthesis minus open parenthesis 8 minus 5 close parenthesis |")[$16 − 2 | 3 ( 1 − 4 ) − ( 8 − 5 ) |$]

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

#math.equation(block: true, alt: "−5 plus 2 open parenthesis −3 close parenthesis squared minus 9")[$−5 + 2 attach(( −3 ), t: 2) − 9$]

#math.equation(block: true, alt: "the fraction 180 over 204")[$frac(180, 204)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 15 over 17")[$frac(15, 17)$]
]

#math.equation(block: true, alt: "minus the fraction 7 over 18 plus the fraction 5 over 12")[$− frac(7, 18) + frac(5, 12)$]

#math.equation(block: true, alt: "the fraction 4 over 5 divided by open parenthesis minus the fraction 12 over 25 close parenthesis")[$frac(4, 5) ÷ ( − frac(12, 25) )$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 5 over 3")[$− frac(5, 3)$]
]

#math.equation(block: true, alt: "the fraction 9 minus 3 times 9 over 15 minus 9")[$frac(9 − 3 · 9, 15 − 9)$]

#math.equation(block: true, alt: "the fraction 4 open parenthesis −3 plus 2 open parenthesis 3 minus 6 close parenthesis close parenthesis over 3 open parenthesis 11 minus 3 open parenthesis 2 plus 3 close parenthesis close parenthesis")[$frac(4 ( −3 + 2 ( 3 − 6 ) ), 3 ( 11 − 3 ( 2 + 3 ) ))$]

#solutionbox[
#math.equation(block: true, alt: "3")[$3$]
]

#math.equation(block: true, alt: "the fraction 5 over 13 times 47 times the fraction 13 over 5")[$frac(5, 13) · 47 · frac(13, 5)$]

#math.equation(block: true, alt: "the fraction minus the fraction 5 over 9 over the fraction 10 over 21")[$frac(− frac(5, 9), frac(10, 21))$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 7 over 6")[$− frac(7, 6)$]
]

#math.equation(block: true, alt: "−4.8 plus open parenthesis −6.7 close parenthesis")[$−4.8 + ( −6.7 )$]

#math.equation(block: true, alt: "34.6 minus 100")[$34.6 − 100$]

#solutionbox[
#math.equation(block: true, alt: "−65.4")[$−65.4$]
]

#math.equation(block: true, alt: "−12.04 times open parenthesis 4.2 close parenthesis")[$−12.04 · ( 4.2 )$]

#math.equation(block: true, alt: "−8 divided by 0.05")[$−8 ÷ 0.05$]

#solutionbox[
−160
]

#math.equation(block: true, alt: "the square root of −121")[$sqrt(−121)$]

#math.equation(block: true, alt: "open parenthesis the fraction 8 over 13 plus the fraction 5 over 7 close parenthesis plus the fraction 2 over 7")[$( frac(8, 13) + frac(5, 7) ) + frac(2, 7)$]

#solutionbox[
#math.equation(block: true, alt: "1 the fraction 8 over 13")[$1 frac(8, 13)$]
]

#math.equation(block: true, alt: "5 x plus open parenthesis −8 y close parenthesis minus 6 x plus 3 y")[$5 x + ( −8 y ) − 6 x + 3 y$]

ⓐ #math.equation(block: false, alt: "the fraction 0 over 9")[$frac(0, 9)$] ⓑ #math.equation(block: false, alt: "the fraction 11 over 0")[$frac(11, 0)$]

#solutionbox[
ⓐ 0 ⓑ undefined
]

#math.equation(block: true, alt: "−3 open parenthesis 8 x minus 5 close parenthesis")[$−3 ( 8 x − 5 )$]

#math.equation(block: true, alt: "6 open parenthesis 3 y minus 1 close parenthesis minus open parenthesis 5 y minus 3 close parenthesis")[$6 ( 3 y − 1 ) − ( 5 y − 3 )$]

#solutionbox[
#math.equation(block: true, alt: "13 y minus 3")[$13 y − 3$]
]