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Equations with Decimals, Fractions, and Parentheses

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Equations with Decimals, Fractions and Parentheses

Pens are $9 per dozen and pencils are $6 per dozen. Janet needs to buy a half dozen of each for school. How much is the total cost of her purchase?

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Khan Academy Solving Equations with the Distributive Property

Guidance

Recall that the distributive property is a mathematical way of grouping terms. It states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. Here, you will use the distributive property with equations that contain decimals and/or fractions. The same rules apply. If the equation has parentheses, your first step is to multiply what is on the outside of the parentheses by what is on the inside of the parentheses. After you remove parentheses, you then solve the equation by combining like terms, moving constants to one side of the equals sign and variables to the other side of the equals sign, and finally isolating the variable to find the solution.

Example A

Solve: \frac{2}{5}(d+4) = 6 .

Solution:

\frac{2}{5}(d+4) &= 6\\\frac{2}{5}d+\frac{8}{5} &= 6 && (\text{Apply the distributive property to remove the parentheses})

Find the LCD for 5, 5, and 1. Since it is 5, multiply the last number by \frac{5}{5} , to get the same denominator.

\frac{2}{5}d+\frac{8}{5} &= \left({\color{red}\frac{5}{5}}\right)6\\\frac{2}{5}d+\frac{8}{5} &= \frac{30}{5} && (\text{Simplify})

Since all of the denominators are the same, the equation becomes:

2d+8 &= 30\\2d+8 {\color{red}-8} &= 30 {\color{red}-8} && (\text{Subtract} \ 8 \ \text{from both sides of the equals sign to isolate the variable})\\2d &= 22 && (\text{Simplify})\\\frac{2d}{{\color{red}2}}&=\frac{22}{{\color{red}2}} && (\text{Divide by} \ 2 \ \text{to solve for the variable})\\d &= 11

Therefore d = 11 .

\text{Check:} &\\\frac{2}{5}(d+4) &= 6\\\frac{2}{5} ({\color{red}11} +4) &=6\\\frac{2}{5}(15) &= 6\\\frac{30}{5} &= 6\\6 &= 6 \ \

Example B

Solve: \frac{1}{4}(3x+7) =2 .

Solution:

\frac{1}{4}(3x+7) &= -2\\\frac{3}{4}x+\frac{7}{4} &= -2 && (\text{Apply the distributive property to remove the parentheses})

Find the LCD for 4, 4, and 1. Since it is 4, multiply the last number by \frac{4}{4} , to get the same denominator.

\frac{3}{4}x+\frac{7}{4} &= \left({\color{red}\frac{4}{4}}\right)-2\\\frac{3}{4}x+\frac{7}{4} &= \frac{-8}{4} && (\text{Simplify})

Since all of the denominators are the same, the equation becomes:

3x+7 &= -8\\3x+7 {\color{red}-7} &= -8 {\color{red}-7} && (\text{Subtract} \ 7 \ \text{from both sides of the equals sign to isolate the variable})\\3x &= -15 && (\text{Simplify})\\\frac{3x}{{\color{red}3}} &= \frac{-15}{{\color{red}3}} && (\text{Divide by} \ 3 \ \text{to solve for the variable})\\x &= -5

Therefore x = -5 .

\text{Check:} &\\\frac{1}{4}(3x+7) &= -2\\\frac{1}{4} (3 ({\color{red}-5})+7) &= -2\\\frac{1}{4}(-15+7) &= -2\\\frac{1}{4}(-8) &= -2\\\frac{-8}{4} &= -2\\-2 &= -2 \ \

Example C

Solve: \frac{1}{3}(x-2) = -\frac{2}{3}(2x+4) .

Solution:

\frac{1}{3}(x-2) &= -\frac{2}{3}(2x+4)\\\frac{1}{3}x-\frac{2}{3} &= -\frac{4}{3}x-\frac{8}{3} && (\text{Apply the distributive property to remove the parentheses})

Since all of the denominators are the same, the equation becomes:

x-2 &= -4x-8\\x {\color{red}+4x} -2 &= -4x {\color{red}+4x}-8 && (\text{Add} \ 4x \ \text{to both sides of the equals sign to combine variables})\\5x-2 &= -8 && (\text{Simplify})\\5x-2 {\color{red}+2} &= -8 {\color{red}+2} && (\text{Add} \ 2 \ \text{to both sides of the equation to isolate the variable})\\5x &= -6 && (\text{Simplify})\\\frac{5x}{{\color{red}5}} &= \frac{-6}{{\color{red}5}} && (\text{Divide both sides by} \ 5 \ \text{to solve for the variable})\\x&=\frac{-6}{5} && (\text{Simplify})

Therefore x=\frac{-6}{5} .

\text{Check:}&\\\frac{1}{3}(x-2) &= -\frac{2}{3}(2x+4)\\\frac{1}{3} \left( {\color{red}\left(\frac{-6}{5}\right)} -2 \right) &= -\frac{2}{3} \left(2 {\color{red}\left(\frac{-6}{5}\right)}+4\right)\\0.33(-1.2-2) &= -0.67(2(-1.2)+4)\\0.33(-3.2) &= -0.67(1.6)\\-1.1 &= -1.1 \ \

Concept Problem Revisited

Pens are $9 per dozen and pencils are $6 per dozen. Janet needs to buy a half dozen of each for school. How much is the total cost of her purchase?

First you should write down what you know:

Let x = total cost

Cost of pens: $9/dozen

Cost of pencils: $6/dozen

Janet needs one half dozen of each.

The total cost would therefore be:

\frac{1}{2}(\$9+\$6) &= x\\\frac{\$ 9}{2}+\frac{\$ 6}{2} &= x\\\$4.50+\$3.00 &= x\\\$7.50 &= x

Therefore Janet would need $7.50 to buy these supplies.

Vocabulary

Distributive Property
The distributive property is a mathematical way of grouping terms. It states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, in the expression: {\color{red}\frac{2}{3}} ({\color{blue}x + 5}) , the distributive property states that the product of a number ({\color{red}\frac{2}{3}}) and a sum ({\color{blue}x + 5}) is equal to the sum of the individual products of the number ({\color{red}\frac{2}{3}}) and the addends ({\color{blue}x} and {\color{blue}5}) .

Guided Practice

1. Solve for x: \frac{1}{2}(5x+3)=1 .

2. Solve for x: \frac{2}{3}(9x-6)=2 .

3. Solve for x: \frac{2}{3}(3x+9)=\frac{1}{4}(2x+5) .

Answers:

1.

\frac{1}{2} (5x+3) &=1 \\\frac{5}{2}x+\frac{3}{2}&=1 && (\text{Apply the distributive property to remove the parentheses})

Find the LCD for 2, 2, and 1. Since it is 2, multiply the last number by \frac{2}{2} , to get the same denominator.

\frac{5}{2}x+\frac{3}{2} &= 1 {\color{red}\left(\frac{2}{2}\right)}\\\frac{5}{2}x+\frac{3}{2} &= \frac{2}{2} && (\text{Simplify})

Since all of the denominators are the same, the equation becomes:

5x+3 &= 2\\5x+3 {\color{red}-3} &= 2 {\color{red}-3} && (\text{Subtract} \ 3 \ \text{from both sides of the equals sign to isolate the variable})\\5x &= -1 && (\text{Simplify})\\\frac{5x}{{\color{red}5}} &= \frac{-1}{{\color{red}5}} && (\text{Divide both sides by the} \ 5 \ \text{to solve for the variable})\\x &= \frac{-1}{5} && (\text{Simplify})

Therefore x=\frac{-1}{5} .

2.

\frac{2}{3} (9x-6) &= 2\\\frac{18}{3}x-\frac{12}{3} &= 2 && (\text{Apply the distributive property to remove the parentheses})

Find the LCD for 3, 3, and 1. Since it is 3, multiply the last number by \frac{3}{3} , to get the same denominator.

\frac{18}{3}x-\frac{12}{3} &= 2 {\color{red}\left(\frac{3}{3}\right)}\\\frac{18}{3}x-\frac{12}{3} &= \frac{6}{3} && (\text{Simplify})

Since all of the denominators are the same, the equation becomes:

18x-12 &= 6\\18x-12 {\color{red}+12} &= 6 {\color{red}+12} && (\text{Add} \ 12 \ \text{to both sides of the equals sign to isolate the variable})\\18x &= 18 && (\text{Simplify})\\\frac{18x}{{\color{red}18}} &= \frac{18}{{\color{red}18}} && (\text{Divide both sides by the} \ 18 \ \text{to solve for the variable})\\x &= 1 && (\text{Simplify})

Therefore x=1 .

3.

\frac{2}{3}(3x+9) &= \frac{1}{4}(2x+5)\\\frac{6}{3}x+\frac{18}{3} &= \frac{2}{4}x+\frac{5}{4} && (\text{Apply the distributive property to remove the parentheses})

Find the LCD for 3, 3, and 4, 4. Since it is 12, multiply the first two fractions by \frac{4}{4} and the second two fractions by \frac{3}{3} , to get the same denominator.

\left({\color{red}\frac{4}{4}}\right) \frac{6}{3}x+\left({\color{red}\frac{4}{4}}\right) \frac{18}{3} &= \left({\color{red}\frac{3}{3}}\right) \frac{2}{4}x+\left({\color{red}\frac{3}{3}}\right) \frac{5}{4}\\\frac{24}{12}x+\frac{72}{12} &= \frac{6}{12}x+\frac{15}{12} && (\text{Simplify})

Since all of the denominators are the same, the equation becomes:

24x+72 &= 6x+15\\24x+72 {\color{red}-72} &= 6x+15 {\color{red}-72} && (\text{Subtract} \ 72 \ \text{from both sides of the equals sign to isolate the variable})\\24x &= 6x-57 && (\text{Simplify})\\24x {\color{red}-6x} &= 6x {\color{red}-6x} - 57 && (\text{Subtract} \ 6x \ \text{from both sides of the equals sign to get variables on same side})\\18x &= -57 && (\text{Simplify})\\\frac{18x}{{\color{red}18}} &= \frac{-57}{{\color{red}18}} && (\text{Divide both sides by} \ 18 \ \text{to solve for the variable})\\x &= \frac{-57}{18} && (\text{Simplify})

Practice

Solve for the variable in each of the following equations.

  1. \frac{1}{2} (x+5)=6
  2. \frac{1}{4}(g+2)=8
  3. 0.4(b+2)=2
  4. 0.5(r-12)=4
  5. \frac{1}{4}(x-16)=7
  1. 26.5-k=0.5(50-k)
  2. 2(1.5c+4)=-1
  3. -\frac{1}{2}(3x-5)=7
  4. 0.35+0.10(m-1)=5.45
  5. \frac{1}{4}+\frac{2}{3}(t+1)=\frac{1}{2}
  1. \frac{1}{2}x-3 (x+4)=\frac{2}{3}
  2. -\frac{5}{8}x+x=\frac{1}{8}
  3. 0.4(12-d)=18
  4. 0.25(x+3)=0.4(x-5)
  5. \frac{2}{3}(t-2)=\frac{3}{4}(t+2)

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