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== 7.3#h(0.6em)Distributive Property

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

Multiply: #math.equation(block: false, alt: "3 open parenthesis 0.25 close parenthesis .")[$3 ( 0.25 ) .$] #linebreak() 
If you missed this problem, review

#solutionbox[
#math.equation(block: true, alt: "0.75")[$0.75$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "10 minus open parenthesis −2 close parenthesis open parenthesis 3 close parenthesis .")[$10 − ( −2 ) ( 3 ) .$] #linebreak() 
If you missed this problem, review

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Combine like terms: #math.equation(block: false, alt: "9 y plus 17 plus 3 y minus 2 .")[$9 y + 17 + 3 y − 2 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "12 y plus 15")[$12 y + 15$]
]
]


=== Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need #math.equation(block: false, alt: "$9.25 ;")[$"$9.25" ;$] that is, #math.equation(block: false, alt: "9")[$9$] dollars and #math.equation(block: false, alt: "1")[$1$] quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

#figure(figph[The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills.], alt: "The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills.", caption: none)

#figure(figph[The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.], alt: "The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.", caption: none)

They need #math.equation(block: false, alt: "3")[$3$] times #math.equation(block: false, alt: "$9 ,")[$"$9" ,$] so #math.equation(block: false, alt: "$27 ,")[$"$27" ,$] and #math.equation(block: false, alt: "3")[$3$] times #math.equation(block: false, alt: "1")[$1$] quarter, so #math.equation(block: false, alt: "75")[$75$] cents. In total, they need #math.equation(block: false, alt: "$27.75 .")[$"$27.75" .$]

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]

If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are real numbers, then

#math.equation(block: true, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$]
]

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

#math.equation(block: true, alt: "3 open parenthesis 9.25 close parenthesis; 3 open parenthesis 9, plus, 0.25 close parenthesis; 3 open parenthesis 9 close parenthesis, plus, 3 open parenthesis 0.25 close parenthesis; 27, plus, 0.75; 27.75")[$& 3 ( 9.25 ) &  \ 3 ( 9 & + & 0.25 ) \ 3 ( 9 ) & + & 3 ( 0.25 ) \ 27 & + & 0.75 \  & 27.75 &$]In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression #math.equation(block: false, alt: "3 open parenthesis x plus 4 close parenthesis ,")[$3 ( x + 4 ) ,$] the order of operations says to work in the parentheses first. But we cannot add #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "4 ,")[$4 ,$] since they are not like terms. So we use the Distributive Property, as shown.

#examplebox("Example 1")[][
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[
4#emph[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[
6#emph[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 image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.], alt: "The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.", caption: none)

#figure(figph[The image shows and equation. On the left side of the equation is the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. This is set equal to 3 times x plus 3 times 4.], alt: "The image shows and equation. On the left side of the equation is the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. This is set equal to 3 times x plus 3 times 4.", caption: none)

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "6 open parenthesis 5 y plus 1 close parenthesis .")[$6 ( 5 y + 1 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression 6(5y + 1) with two blue arrows showing the distribution of 6 to both 5y and 1, illustrating the distributive property.], alt: "A mathematical expression 6(5y + 1) with two blue arrows showing the distribution of 6 to both 5y and 1, illustrating the distributive property.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression showing 6 multiplied by 5y, plus 6 multiplied by 1.], alt: "A mathematical expression showing 6 multiplied by 5y, plus 6 multiplied by 1.", caption: none)],
  [Multiply.], [#figure(figph[The image shows the mathematical expression '30y + 6' in black text on a white background. The numbers '30' and '6' are visible, as is the variable 'y' and the plus sign '+'.], alt: "The image shows the mathematical expression '30y + 6' in black text on a white background. The numbers '30' and '6' are visible, as is the variable 'y' and the plus sign '+'.", caption: none)],
))
]
]

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

#solutionbox[
27#emph[y] + 72
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 open parenthesis 5 w plus 9 close parenthesis .")[$5 ( 5 w + 9 ) .$]

#solutionbox[
25#emph[w] + 45
]
]

The distributive property can be used to simplify expressions that look slightly different from #math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis .")[$a ( b + c ) .$] Here are two other forms.

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

If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are real numbers, then

#math.equation(block: true, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$]Other forms

#math.equation(block: true, alt: "a open parenthesis b minus c close parenthesis equals a b minus a c")[$a ( b − c ) = a b − a c$]#math.equation(block: true, alt: "open parenthesis b plus c close parenthesis a equals b a plus c a")[$( b + c ) a = b a + c a$]
]

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An image illustrating the distributive property in algebra, showing the expression 2(x - 3) with two blue arrows indicating that 2 should be multiplied by both x and -3.], alt: "An image illustrating the distributive property in algebra, showing the expression 2(x - 3) with two blue arrows indicating that 2 should be multiplied by both x and -3.", caption: none)]),
  [Distribute.], [#figure(figph[The image shows the mathematical expression 2 multiplied by x minus 2 multiplied by 3.], alt: "The image shows the mathematical expression 2 multiplied by x minus 2 multiplied by 3.", caption: none)],
  [Multiply.], [#figure(figph[The image shows the mathematical expression '2x - 6' in bold, black text on a white background. The terms '2x' and '6' are separated by a minus sign.], alt: "The image shows the mathematical expression '2x - 6' in bold, black text on a white background. The terms '2x' and '6' are separated by a minus sign.", caption: none)],
))
]
]

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

#solutionbox[
7#emph[x] − 42
]
]

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

#solutionbox[
8#emph[x] − 40
]
]

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Illustration of the distributive property for the expression (3/4)(n + 12), with light blue arrows showing 3/4 being multiplied by 'n' and by '12'.], alt: "Illustration of the distributive property for the expression (3/4)(n + 12), with light blue arrows showing 3/4 being multiplied by 'n' and by '12'.", caption: none)]),
  [Distribute.], [#figure(figph[The image shows the mathematical expression three-fourths times n plus three-fourths times twelve, written as (3/4) \* n + (3/4) \* 12.], alt: "The image shows the mathematical expression three-fourths times n plus three-fourths times twelve, written as (3/4) * n + (3/4) * 12.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression displays three-fourths multiplied by 'n', with the result then added to nine, set against a plain white background.], alt: "A mathematical expression displays three-fourths multiplied by 'n', with the result then added to nine, set against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 5 open parenthesis p plus 10 close parenthesis .")[$frac(2, 5) ( p + 10 ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 over 7 open parenthesis u plus 21 close parenthesis .")[$frac(3, 7) ( u + 21 ) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 7 u plus 9")[$frac(3, 7) u + 9$]
]
]

#examplebox("Example 5")[][
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[An image illustrating the distributive property, showing how to multiply 8 by each term inside the parentheses (3/8x and 1/4) in the expression 8(3/8x + 1/4).], alt: "An image illustrating the distributive property, showing how to multiply 8 by each term inside the parentheses (3/8x and 1/4) in the expression 8(3/8x + 1/4).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression shows 8 multiplied by the fraction 3/8 times x, plus 8 multiplied by the fraction 1/4.], alt: "A mathematical expression shows 8 multiplied by the fraction 3/8 times x, plus 8 multiplied by the fraction 1/4.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the algebraic expression '3x + 2' in black characters against a white background.], alt: "The image displays the algebraic expression '3x + 2' in black characters 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[
5#emph[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[
4#emph[n] + 9
]
]

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

#examplebox("Example 6")[][
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[Illustration of the distributive property with the expression 100(0.3 + 0.25q). Arrows show 100 multiplying 0.3 and 0.25q.], alt: "Illustration of the distributive property with the expression 100(0.3 + 0.25q). Arrows show 100 multiplying 0.3 and 0.25q.", caption: none)]),
  [Distribute.], [#figure(figph[An algebraic expression is shown, which reads as one hundred multiplied by zero point three, plus one hundred multiplied by zero point two five q.], alt: "An algebraic expression is shown, which reads as one hundred multiplied by zero point three, plus one hundred multiplied by zero point two five q.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the mathematical expression '30 + 25q' in a clear, black font against a white background.], alt: "The image displays the mathematical expression '30 + 25q' in a clear, black font against a white background.", 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[
70 + 15#emph[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[
4 + 35#emph[d]
]
]

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the algebraic expression m(n-4) with two blue arrows demonstrating the distributive property by indicating that 'm' multiplies both 'n' and '-4' within the parentheses.], alt: "The image shows the algebraic expression m(n-4) with two blue arrows demonstrating the distributive property by indicating that 'm' multiplies both 'n' and '-4' within the parentheses.", caption: none)]),
  [Distribute.], [#figure(figph[The mathematical expression m multiplied by n, minus m multiplied by 4, is displayed in bold black font on a white background.], alt: "The mathematical expression m multiplied by n, minus m multiplied by 4, is displayed in bold black font on a white background.", caption: none)],
  [Multiply.], [#figure(figph[The image shows the mathematical expression 'mn - 4m' in a bold, italicized font on a white background.], alt: "The image shows the mathematical expression 'mn - 4m' in a bold, italicized font on a white background.", caption: none)],
))

Notice that we wrote #math.equation(block: false, alt: "m times 4 as 4 m .")[$m · 4 #h(0.2em) "as" #h(0.2em) 4 m .$] We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.
]
]

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

#solutionbox[
#emph[rs] − 2#emph[r]
]
]

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

#solutionbox[
#emph[yz] − 8#emph[y]
]
]

The next example will use the ‘backwards’ form of the Distributive Property, #math.equation(block: false, alt: "open parenthesis b plus c close parenthesis a equals b a plus c a .")[$( b + c ) a = b a + c a .$]

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic expression (x + 8)p with arrows illustrating the distributive property, showing 'p' multiplying both 'x' and '8' within the parentheses.], alt: "An algebraic expression (x + 8)p with arrows illustrating the distributive property, showing 'p' multiplying both 'x' and '8' within the parentheses.", caption: none)]),
  [Distribute.], [#figure(figph[The mathematical expression 'px + 8p' is displayed in bold black text on a white background.], alt: "The mathematical expression 'px + 8p' is displayed in bold black text on a white background.", caption: none)],
))
]
]

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

#solutionbox[
#emph[xp] + 2#emph[p]
]
]

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

#solutionbox[
#emph[yq] + 4#emph[q]
]
]

When you distribute a negative number, you need to be extra careful to get the signs correct.

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "−2 open parenthesis 4 y plus 1 close parenthesis .")[$−2 ( 4 y + 1 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the algebraic expression -2(4y + 1) with blue arrows illustrating the distributive property. The arrows indicate that -2 should be multiplied by both 4y and 1 inside the parentheses.], alt: "The image displays the algebraic expression -2(4y + 1) with blue arrows illustrating the distributive property. The arrows indicate that -2 should be multiplied by both 4y and 1 inside the parentheses.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression shows the sum of two products: -2 multiplied by 4y, and -2 multiplied by 1. It is written as -2   4y + (-2)   1.], alt: "A mathematical expression shows the sum of two products: -2 multiplied by 4y, and -2 multiplied by 1. It is written as -2 4y + (-2) 1.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression '-8y - 2' is displayed in a dark gray font against a white background.], alt: "The mathematical expression '-8y - 2' is displayed in a dark gray font against a white background.", caption: none)],
))
]
]

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

#solutionbox[
−18#emph[m] − 15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−6 open parenthesis 8 n plus 11 close parenthesis .")[$−6 ( 8 n + 11 ) .$]

#solutionbox[
−48#emph[n] − 66
]
]

#examplebox("Example 10")[][
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([], [#figure(figph[A mathematical expression shows -11 multiplied by the quantity (4 - 3a).], alt: "A mathematical expression shows -11 multiplied by the quantity (4 - 3a).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression showing the subtraction of two terms, where the first term is -11 multiplied by 4, and the second term is -11 multiplied by 3a. The expression is -11 \* 4 - (-11) \* 3a.], alt: "A mathematical expression showing the subtraction of two terms, where the first term is -11 multiplied by 4, and the second term is -11 multiplied by 3a. The expression is -11 * 4 - (-11) * 3a.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the mathematical expression '-44 - (-33a)' on a white background, demonstrating subtraction involving a negative number and a variable term enclosed in parentheses.], alt: "The image displays the mathematical expression '-44 - (-33a)' on a white background, demonstrating subtraction involving a negative number and a variable term enclosed in parentheses.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression -44 + 33a is shown on a white background.], alt: "The mathematical expression -44 + 33a is shown on a white background.", caption: none)],
))

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[
−10 + 15#emph[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[
−56 + 105#emph[y]
]
]

In the next example, we will show how to use the #strong[Distributive Property] to find the opposite of an expression. Remember, #math.equation(block: false, alt: "minus a equals −1 times a .")[$− a = −1 · a .$]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "minus open parenthesis y plus 5 close parenthesis .")[$− ( y + 5 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression shows the negative of the sum of 'y' and '5', written as -(y+5).], alt: "A mathematical expression shows the negative of the sum of 'y' and '5', written as -(y+5).", caption: none)]),
  [Multiplying by −1 results in the opposite.], [#figure(figph[The mathematical expression -1(y + 5) is displayed in black text on a white background, representing a negative one multiplied by the sum of y and five.], alt: "The mathematical expression -1(y + 5) is displayed in black text on a white background, representing a negative one multiplied by the sum of y and five.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical expression showing the term -1 multiplied by 'y', added to the product of -1 and 5. The full expression is -1\*y + (-1)\*5.], alt: "A mathematical expression showing the term -1 multiplied by 'y', added to the product of -1 and 5. The full expression is -1*y + (-1)*5.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical expression '-y + (-5)', which represents the addition of negative y and negative 5.], alt: "The image shows the mathematical expression '-y + (-5)', which represents the addition of negative y and negative 5.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression -y - 5 is displayed in black characters on a white background.], alt: "The mathematical expression -y - 5 is displayed in black characters on a white background.", caption: none)],
))
]
]

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

#solutionbox[
−#emph[z] + 11
]
]

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

#solutionbox[
−#emph[x] + 4
]
]

Sometimes we 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 12")[][
Simplify: #math.equation(block: false, alt: "8 minus 2 open parenthesis x plus 3 close parenthesis .")[$8 − 2 ( x + 3 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the mathematical expression 8 - 2(x + 3).], alt: "The image shows the mathematical expression 8 - 2(x + 3).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical expression reads 8 minus 2 multiplied by x, minus 2 multiplied by 3.], alt: "A mathematical expression reads 8 minus 2 multiplied by x, minus 2 multiplied by 3.", caption: none)],
  [Multiply.], [#figure(figph[The image shows the mathematical expression 8 - 2x - 6 in a clear, dark font against a white background.], alt: "The image shows the mathematical expression 8 - 2x - 6 in a clear, dark font against a white background.", caption: none)],
  [Combine like terms.], [#figure(figph[The mathematical expression -2x + 2 is shown on a white background. The text is rendered in a clear, bold black font, typical of mathematical notation.], alt: "The mathematical expression -2x + 2 is shown on a white background. The text is rendered in a clear, bold black font, typical of mathematical notation.", caption: none)],
))
]
]

#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[
−3#emph[x] + 3
]
]

#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[
2#emph[x] − 20
]
]

#examplebox("Example 13")[][
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([], [#figure(figph[A mathematical expression featuring the terms 4(x - 8) - (x + 3) is displayed on a white background.], alt: "A mathematical expression featuring the terms 4(x - 8) - (x + 3) is displayed on a white background.", caption: none)]),
  [Distribute.], [#figure(figph[The image shows a mathematical expression: 4x - 32 - x - 3. It displays a series of numbers and variables connected by subtraction signs, all in a black font against a white background.], alt: "The image shows a mathematical expression: 4x - 32 - x - 3. It displays a series of numbers and variables connected by subtraction signs, all in a black font against a white background.", caption: none)],
  [Combine like terms.], [#figure(figph[The image shows the mathematical expression '3x - 35' in black text on a white background.], alt: "The image shows the mathematical expression '3x - 35' in black text on a white background.", caption: none)],
))
]
]

#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[
5#emph[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[
7#emph[x] − 13
]
]


=== Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the #strong[Distributive Property] always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

#examplebox("Example 14")[][
When #math.equation(block: false, alt: "y equals 10")[$y = 10$] evaluate:  ⓐ #math.equation(block: false, alt: "6 open parenthesis 5 y plus 1 close parenthesis")[$#h(0.2em) 6 ( 5 y + 1 )$] ⓑ #math.equation(block: false, alt: "6 times 5 y plus 6 times 1 .")[$#h(0.2em) 6 · 5 y + 6 · 1 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "6 open parenthesis 5 y plus 1 close parenthesis")[$6 ( 5 y + 1 )$]],
  [#figure(figph[The text reads 'Substitute 10 for y.', with '10' highlighted in red.], alt: "The text reads 'Substitute 10 for y.', with '10' highlighted in red.", caption: none)], [#figure(figph[A mathematical expression 6(5 ⋅ 10 + 1) is displayed in black font, with the number 10 highlighted in red, on a white background. It represents a calculation following the order of operations.], alt: "A mathematical expression 6(5 ⋅ 10 + 1) is displayed in black font, with the number 10 highlighted in red, on a white background. It represents a calculation following the order of operations.", caption: none)],
  [Simplify in the parentheses.], [#math.equation(block: false, alt: "6 open parenthesis 51 close parenthesis")[$6 ( 51 )$]],
  [Multiply.], [#math.equation(block: false, alt: "306")[$306$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[A mathematical expression showing the distributive property: 6 multiplied by 5y, plus 6 multiplied by 1. The expression is 6 \* 5y + 6 \* 1, rendered in a black font on a white background.], alt: "A mathematical expression showing the distributive property: 6 multiplied by 5y, plus 6 multiplied by 1. The expression is 6 * 5y + 6 * 1, rendered in a black font on a white background.", caption: none)],
  [#figure(figph[The image displays the instruction 'Substitute 10 for y.' written in a blue-green font, with the number '10' highlighted in red, all against a plain white background.], alt: "The image displays the instruction 'Substitute 10 for y.' written in a blue-green font, with the number '10' highlighted in red, all against a plain white background.", caption: none)], [#figure(figph[A mathematical expression reads '6 times 5 times 10 plus 6 times 1', with the number 10 highlighted in red.], alt: "A mathematical expression reads '6 times 5 times 10 plus 6 times 1', with the number 10 highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical expression '300 + 6' in black text against a white background.], alt: "The image shows the mathematical expression '300 + 6' in black text against a white background.", caption: none)],
  [Add.], [#figure(figph[The number '306' is displayed in black text on a plain white background.], alt: "The number '306' is displayed in black text on a plain white background.", caption: none)],
))

Notice, the answers are the same. When #math.equation(block: false, alt: "y equals 10 ,")[$y = 10 ,$]

#math.equation(block: true, alt: "6 open parenthesis 5 y plus 1 close parenthesis equals 6 times 5 y plus 6 times 1 .")[$6 ( 5 y + 1 ) = 6 · 5 y + 6 · 1 .$]Try it yourself for a different value of #math.equation(block: false, alt: "y .")[$y .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "w equals 3 :")[$w = 3 ":" #h(0.2em)$]ⓐ #math.equation(block: false, alt: "5 open parenthesis 5 w plus 9 close parenthesis")[$#h(0.2em) 5 ( 5 w + 9 ) #h(0.2em)$]ⓑ #math.equation(block: false, alt: "5 times 5 w plus 5 times 9 .")[$#h(0.2em) 5 · 5 w + 5 · 9 .$]

#solutionbox[
+ ⓐ 120
+ ⓑ 120
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "y equals 2 :")[$y = 2 ":" #h(0.2em)$]ⓐ #math.equation(block: false, alt: "9 open parenthesis 3 y plus 8 close parenthesis")[$#h(0.2em) 9 ( 3 y + 8 ) #h(0.2em)$]ⓑ #math.equation(block: false, alt: "9 times 3 y plus 9 times 8 .")[$#h(0.2em) 9 · 3 y + 9 · 8 .$]

#solutionbox[
+ ⓐ 126
+ ⓑ 126
]
]

#examplebox("Example 15")[][
When #math.equation(block: false, alt: "y equals 3 ,")[$y = 3 ,$] evaluate ⓐ #math.equation(block: false, alt: "−2 open parenthesis 4 y plus 1 close parenthesis")[$#h(0.2em) −2 ( 4 y + 1 )$]ⓑ #math.equation(block: false, alt: "−2 times 4 y plus open parenthesis −2 close parenthesis times 1 .")[$#h(0.2em) −2 · 4 y + ( −2 ) · 1 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "−2 open parenthesis 4 y plus 1 close parenthesis")[$#h(0.2em) −2 ( 4 y + 1 )$]],
  [#figure(figph[The image displays the instruction 'Substitute 3 for y.' in a sans-serif font, with the number 3 highlighted in red, on a plain white background.], alt: "The image displays the instruction 'Substitute 3 for y.' in a sans-serif font, with the number 3 highlighted in red, on a plain white background.", caption: none)], [#figure(figph[A mathematical expression is displayed, showing a negative two multiplied by the sum of four times three and one, enclosed in parentheses: -2(4 \* 3 + 1). The number '3' is highlighted in red.], alt: "A mathematical expression is displayed, showing a negative two multiplied by the sum of four times three and one, enclosed in parentheses: -2(4 * 3 + 1). The number '3' is highlighted in red.", caption: none)],
  [Simplify in the parentheses.], [#math.equation(block: false, alt: "−2 open parenthesis 13 close parenthesis")[$−2 ( 13 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−26")[$−26$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "−2 times 4 y plus open parenthesis −2 close parenthesis times 1")[$#h(0.2em) −2 · 4 y + ( −2 ) · 1$]],
  [#figure(figph[The text reads 'Substitute 3 for y.' in a sans-serif font, with the number '3' highlighted in red and the rest of the text in a dark teal color. The text is centered on a white background.], alt: "The text reads 'Substitute 3 for y.' in a sans-serif font, with the number '3' highlighted in red and the rest of the text in a dark teal color. The text is centered on a white background.", caption: none)], [#figure(figph[A mathematical expression shows the calculation '-2 multiplied by 4 multiplied by 3, plus -2 multiplied by 1', with the number 3 highlighted in red.], alt: "A mathematical expression shows the calculation '-2 multiplied by 4 multiplied by 3, plus -2 multiplied by 1', with the number 3 highlighted in red.", caption: none)],
  [Multiply.], [#math.equation(block: false, alt: "−24 minus 2")[$−24 − 2$]],
  [Subtract.], [#math.equation(block: false, alt: "−26")[$−26$]],
  [The answers are the same. When #math.equation(block: false, alt: "y equals 3 ,")[$y = 3 ,$]], [#math.equation(block: false, alt: "−2 open parenthesis 4 y plus 1 close parenthesis equals −8 y minus 2")[$−2 ( 4 y + 1 ) = −8 y − 2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "n equals −2 :")[$n = −2 ":" #h(0.2em)$]ⓐ #math.equation(block: false, alt: "−6 open parenthesis 8 n plus 11 close parenthesis")[$#h(0.2em) −6 ( 8 n + 11 )$]ⓑ #math.equation(block: false, alt: "−6 times 8 n plus open parenthesis −6 close parenthesis times 11 .")[$#h(0.2em) −6 · 8 n + ( −6 ) · 11 .$]

#solutionbox[
+ ⓐ 30
+ ⓑ 30
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "m equals −1 :")[$m = −1 ":"$]ⓐ #math.equation(block: false, alt: "−3 open parenthesis 6 m plus 5 close parenthesis")[$#h(0.2em) #h(0.2em) −3 ( 6 m + 5 ) #h(0.2em)$]ⓑ #math.equation(block: false, alt: "−3 times 6 m plus open parenthesis −3 close parenthesis times 5 .")[$#h(0.2em) −3 · 6 m + ( −3 ) · 5 .$]

#solutionbox[
+ ⓐ 3
+ ⓑ 3
]
]

#examplebox("Example 16")[][
When #math.equation(block: false, alt: "y equals 35")[$y = 35$] evaluate ⓐ #math.equation(block: false, alt: "− open parenthesis y plus 5 close parenthesis")[$#h(0.2em) "−" ( y + 5 )$] and ⓑ #math.equation(block: false, alt: "− y minus 5")[$#h(0.2em) "−" "y" − 5$] to show that #math.equation(block: false, alt: "minus open parenthesis y plus 5 close parenthesis equals − y minus 5 .")[$− ( y + 5 ) = "−" "y" − 5 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "− open parenthesis y plus 5 close parenthesis")[$#h(0.2em) "−" ( y + 5 )$]],
  [#figure(figph[The image displays the text], alt: "The image displays the text", caption: none)], [#figure(figph[The image displays the mathematical expression -(35 + 5), where 35 is highlighted in red, signifying that 35 and 5 are enclosed within parentheses and preceded by a negative sign.], alt: "The image displays the mathematical expression -(35 + 5), where 35 is highlighted in red, signifying that 35 and 5 are enclosed within parentheses and preceded by a negative sign.", caption: none)],
  [Add in the parentheses.], [#math.equation(block: false, alt: "minus open parenthesis 40 close parenthesis")[$− ( 40 )$]],
  [Simplify.], [#math.equation(block: false, alt: "−40")[$−40$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "minus y minus 5")[$#h(0.2em) − y − 5$]],
  [#figure(figph[The text reads 'Substitute 35 for y.' with the number 35 highlighted in red.], alt: "The text reads 'Substitute 35 for y.' with the number 35 highlighted in red.", caption: none)], [#figure(figph[A mathematical expression showing '-35 - -5' with '-35' in red and '- -5' in black, illustrating subtraction of a negative number.], alt: "A mathematical expression showing '-35 - -5' with '-35' in red and '- -5' in black, illustrating subtraction of a negative number.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "−40")[$−40$]],
  [The answers are the same when #math.equation(block: false, alt: "y equals 35 ,")[$y = 35 ,$] demonstrating that], [#math.equation(block: false, alt: "minus open parenthesis y plus 5 close parenthesis equals − y minus 5")[$− ( y + 5 ) = "−" y − 5$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "x equals 36 :")[$x = 36 ":"$]ⓐ #math.equation(block: false, alt: "minus open parenthesis x minus 4 close parenthesis")[$#h(0.2em) #h(0.2em) − ( x − 4 )$]ⓑ #math.equation(block: false, alt: "minus x plus 4")[$#h(0.2em) − x + 4$] to show that #math.equation(block: false, alt: "minus open parenthesis x minus 4 close parenthesis equals minus x plus 4 .")[$− ( x − 4 ) = − x + 4 .$]

#solutionbox[
+ ⓐ −32
+ ⓑ −32
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate when #math.equation(block: false, alt: "z equals 55 :")[$z = 55 ":"$]ⓐ #math.equation(block: false, alt: "minus open parenthesis z minus 10 close parenthesis")[$#h(0.2em) #h(0.2em) − ( z − 10 )$]ⓑ #math.equation(block: false, alt: "minus z plus 10")[$#h(0.2em) − z + 10$] to show that #math.equation(block: false, alt: "minus open parenthesis z minus 10 close parenthesis equals minus z plus 10 .")[$− ( z − 10 ) = − z + 10 .$]

#solutionbox[
+ ⓐ −45
+ ⓑ −45
]
]

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

- #link("https://www.openstax.org/l/24ModelDist")[Model Distribution]
- #link("https://www.openstax.org/l/24DistProp")[The Distributive Property]
]


=== Key Concepts

- #emph[Distributive Property:]- 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")[$a ( b + c ) = a b + a c$]
- #math.equation(block: false, alt: "open parenthesis b plus c close parenthesis a equals b a plus c a")[$( b + c ) a = b a + c a$]
- #math.equation(block: false, alt: "a open parenthesis b minus c close parenthesis equals a b minus a c")[$a ( b − c ) = a b − a c$]


==== Practice Makes Perfect

#strong[Simplify Expressions Using the Distributive Property]

In the following exercises, simplify using the distributive property.

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

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

#solutionbox[
3#emph[a] + 27
]

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

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

#solutionbox[
27#emph[w] + 63
]

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

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

#solutionbox[
7#emph[y] − 91
]

#math.equation(block: true, alt: "7 open parenthesis 3 p minus 8 close parenthesis")[$7 ( 3 p − 8 )$]

#math.equation(block: true, alt: "5 open parenthesis 7 u minus 4 close parenthesis")[$5 ( 7 u − 4 )$]

#solutionbox[
35#emph[u] − 20
]

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

#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis u plus 9 close parenthesis")[$frac(1, 3) ( u + 9 )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3 u plus 3")[$frac(1, 3) u + 3$]
]

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

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

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

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

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

#solutionbox[
3#emph[x] − 4
]

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

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

#solutionbox[
2 + 9#emph[s]
]

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

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

#solutionbox[
#emph[uv] − 10#emph[u]
]

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

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

#solutionbox[
#emph[ax] + 7#emph[x]
]

#math.equation(block: true, alt: "−2 open parenthesis y plus 13 close parenthesis")[$−2 ( y + 13 )$]

#math.equation(block: true, alt: "−3 open parenthesis a plus 11 close parenthesis")[$−3 ( a + 11 )$]

#solutionbox[
−3#emph[a] − 33
]

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

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

#solutionbox[
−81#emph[a] − 36
]

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

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

#solutionbox[
−4#emph[q] + 28
]

#math.equation(block: true, alt: "−9 open parenthesis 3 a minus 7 close parenthesis")[$−9 ( 3 a − 7 )$]

#math.equation(block: true, alt: "−6 open parenthesis 7 x minus 8 close parenthesis")[$−6 ( 7 x − 8 )$]

#solutionbox[
−42#emph[x] + 48
]

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

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

#solutionbox[
−#emph[q] − 11
]

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

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

#solutionbox[
−5#emph[p] + 4
]

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

#math.equation(block: true, alt: "12 plus 8 open parenthesis u minus 1 close parenthesis")[$12 + 8 ( u − 1 )$]

#solutionbox[
8#emph[u] + 4
]

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

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

#solutionbox[
−4#emph[x] + 10
]

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

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

#solutionbox[
−42#emph[n] + 39
]

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

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

#solutionbox[
−#emph[r] + 15
]

#math.equation(block: true, alt: "−12 minus open parenthesis u plus 10 close parenthesis")[$−12 − ( u + 10 )$]

#math.equation(block: true, alt: "−4 minus open parenthesis c minus 10 close parenthesis")[$−4 − ( c − 10 )$]

#solutionbox[
−#emph[c] + 6
]

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

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

#solutionbox[
3#emph[y] + 1
]

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

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

#solutionbox[
47#emph[u] + 60
]

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

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

#solutionbox[
24#emph[x] + 4
]

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

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

#solutionbox[
6#emph[n] − 72
]

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

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

#solutionbox[
17#emph[n] + 76
]

#strong[Evaluate Expressions Using the Distributive Property]

In the following exercises, evaluate both expressions for the given value.

If #math.equation(block: false, alt: "v equals −2 ,")[$v = −2 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "6 open parenthesis 4 v plus 7 close parenthesis")[$#h(0.2em) 6 ( 4 v + 7 )$]
+ ⓑ #math.equation(block: false, alt: "6 times 4 v plus 6 times 7")[$#h(0.2em) 6 · 4 v + 6 · 7$]

If #math.equation(block: false, alt: "u equals −1 ,")[$u = −1 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "8 open parenthesis 5 u plus 12 close parenthesis")[$#h(0.2em) 8 ( 5 u + 12 )$]
+ ⓑ #math.equation(block: false, alt: "8 times 5 u plus 8 times 12")[$#h(0.2em) 8 · 5 u + 8 · 12$]

#solutionbox[
+ ⓐ 56
+ ⓑ 56
]

If #math.equation(block: false, alt: "n equals the fraction 2 over 3 ,")[$n = frac(2, 3) ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "3 open parenthesis n plus the fraction 5 over 6 close parenthesis")[$#h(0.2em) 3 ( n + frac(5, 6) )$]
+ ⓑ #math.equation(block: false, alt: "3 times n plus 3 times the fraction 5 over 6")[$#h(0.2em) 3 · n + 3 · frac(5, 6)$]

If #math.equation(block: false, alt: "y equals the fraction 3 over 4 ,")[$y = frac(3, 4) ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "4 open parenthesis y plus the fraction 3 over 8 close parenthesis")[$#h(0.2em) 4 ( y + frac(3, 8) )$]
+ ⓑ #math.equation(block: false, alt: "4 times y plus 4 times the fraction 3 over 8")[$#h(0.2em) 4 · y + 4 · frac(3, 8)$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "the fraction 9 over 2")[$#h(0.2em) frac(9, 2)$]
+ ⓑ #math.equation(block: false, alt: "the fraction 9 over 2")[$#h(0.2em) frac(9, 2)$]
]

If #math.equation(block: false, alt: "y equals the fraction 7 over 12 ,")[$y = frac(7, 12) ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "−3 open parenthesis 4 y plus 15 close parenthesis")[$#h(0.2em) −3 ( 4 y + 15 )$]
+ ⓑ #math.equation(block: false, alt: "−3 times 4 y plus open parenthesis −3 close parenthesis times 15")[$#h(0.2em) −3 · 4 y + ( −3 ) · 15$]

If #math.equation(block: false, alt: "p equals the fraction 23 over 30 ,")[$p = frac(23, 30) ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "−6 open parenthesis 5 p plus 11 close parenthesis")[$#h(0.2em) −6 ( 5 p + 11 )$]
+ ⓑ #math.equation(block: false, alt: "−6 times 5 p plus open parenthesis −6 close parenthesis times 11")[$#h(0.2em) −6 · 5 p + ( −6 ) · 11$]

#solutionbox[
+ ⓐ −89
+ ⓑ −89
]

If #math.equation(block: false, alt: "m equals 0.4 ,")[$m = 0.4 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "−10 open parenthesis 3 m minus 0.9 close parenthesis")[$#h(0.2em) −10 ( 3 m − 0.9 )$]
+ ⓑ #math.equation(block: false, alt: "−10 times 3 m minus open parenthesis −10 close parenthesis open parenthesis 0.9 close parenthesis")[$#h(0.2em) −10 · 3 m − ( −10 ) ( 0.9 )$]

If #math.equation(block: false, alt: "n equals 0.75 ,")[$n = 0.75 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "−100 open parenthesis 5 n plus 1.5 close parenthesis")[$#h(0.2em) −100 ( 5 n + 1.5 )$]
+ ⓑ #math.equation(block: false, alt: "−100 times 5 n plus open parenthesis −100 close parenthesis open parenthesis 1.5 close parenthesis")[$#h(0.2em) −100 · 5 n + ( −100 ) ( 1.5 )$]

#solutionbox[
+ ⓐ −525
+ ⓑ −525
]

If #math.equation(block: false, alt: "y equals −25 ,")[$y = −25 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis y minus 25 close parenthesis")[$#h(0.2em) − ( y − 25 )$]
+ ⓑ #math.equation(block: false, alt: "minus y plus 25")[$#h(0.2em) − y + 25$]

If #math.equation(block: false, alt: "w equals −80 ,")[$w = −80 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis w minus 80 close parenthesis")[$#h(0.2em) − ( w − 80 )$]
+ ⓑ #math.equation(block: false, alt: "minus w plus 80")[$#h(0.2em) − w + 80$]

#solutionbox[
+ ⓐ 160
+ ⓑ 160
]

If #math.equation(block: false, alt: "p equals 0.19 ,")[$p = 0.19 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis p plus 0.72 close parenthesis")[$#h(0.2em) − ( p + 0.72 )$]
+ ⓑ #math.equation(block: false, alt: "minus p minus 0.72")[$#h(0.2em) − p − 0.72$]

If #math.equation(block: false, alt: "q equals 0.55 ,")[$q = 0.55 ,$] evaluate

+ ⓐ #math.equation(block: false, alt: "minus open parenthesis q plus 0.48 close parenthesis")[$#h(0.2em) − ( q + 0.48 )$]
+ ⓑ #math.equation(block: false, alt: "minus q minus 0.48")[$#h(0.2em) − q − 0.48$]

#solutionbox[
+ ⓐ −1.03
+ ⓑ −1.03
]


==== Everyday Math

#strong[Buying by the case] Joe can buy his favorite ice tea at a convenience store for #math.equation(block: false, alt: "$1.99")[$"$1.99"$] per bottle. At the grocery store, he can buy a case of #math.equation(block: false, alt: "12")[$12$] bottles for #math.equation(block: false, alt: "$23.88 .")[$"$23.88" .$]

+ ⓐ Use the distributive property to find the cost of #math.equation(block: false, alt: "12")[$12$] bottles bought individually at the convenience store. (Hint: notice that #math.equation(block: false, alt: "$1.99")[$"$1.99"$] is #math.equation(block: false, alt: "$2 minus $0.01 .")[$"$2" − "$0.01" .$])
+ ⓑ Is it a bargain to buy the iced tea at the grocery store by the case?

#strong[Multi-pack purchase] Adele’s shampoo sells for #math.equation(block: false, alt: "$3.97")[$"$3.97"$] per bottle at the drug store. At the warehouse store, the same shampoo is sold as a #math.equation(block: false, alt: "3-pack")[$"3-pack"$] for #math.equation(block: false, alt: "$10.49 .")[$"$10.49" .$]

+ ⓐ Show how you can use the distributive property to find the cost of #math.equation(block: false, alt: "3")[$3$] bottles bought individually at the drug store.
+ ⓑ How much would Adele save by buying the #math.equation(block: false, alt: "3-pack")[$"3-pack"$] at the warehouse store?

#solutionbox[
+ ⓐ 3(4 − 0.03) = 11.91
+ ⓑ \$1.42
]


==== Writing Exercises

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.

Explain how you can multiply #math.equation(block: false, alt: "4 open parenthesis $5.97 close parenthesis")[$4 ( "$5.97" )$] without paper or a 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.

#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[A self-assessment table for students to rate their understanding of simplifying and evaluating expressions using the Distributive Property.], alt: "A self-assessment table for students to rate their understanding of simplifying and evaluating expressions using the Distributive Property.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?