Algebra Examples

$\frac{4 x + 8}{15} - \frac{2 x - 8}{9} = \frac{27 x + 18}{27} - x$

Step 1

Simplify $\frac{4 x + 8}{15} - \frac{2 x - 8}{9}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Factor $4$ out of $4 x + 8$ .

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Step 1.1.1.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) + 8}{15} - \frac{2 x - 8}{9} = \frac{27 x + 18}{27} - x$

Step 1.1.1.2

Factor $4$ out of $8$ .

$\frac{4 x + 4 \cdot 2}{15} - \frac{2 x - 8}{9} = \frac{27 x + 18}{27} - x$

Step 1.1.1.3

Factor $4$ out of $4 x + 4 \cdot 2$ .

$\frac{4 (x + 2)}{15} - \frac{2 x - 8}{9} = \frac{27 x + 18}{27} - x$

Step 1.1.2

Factor $2$ out of $2 x - 8$ .

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Step 1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{4 (x + 2)}{15} - \frac{2 (x) - 8}{9} = \frac{27 x + 18}{27} - x$

Step 1.1.2.2

Factor $2$ out of $- 8$ .

$\frac{4 (x + 2)}{15} - \frac{2 x + 2 \cdot - 4}{9} = \frac{27 x + 18}{27} - x$

Step 1.1.2.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

$\frac{4 (x + 2)}{15} - \frac{2 (x - 4)}{9} = \frac{27 x + 18}{27} - x$

Step 1.2

To write $\frac{4 (x + 2)}{15}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 (x + 2)}{15} \cdot \frac{3}{3} - \frac{2 (x - 4)}{9} = \frac{27 x + 18}{27} - x$

Step 1.3

To write $- \frac{2 (x - 4)}{9}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 (x + 2)}{15} \cdot \frac{3}{3} - \frac{2 (x - 4)}{9} \cdot \frac{5}{5} = \frac{27 x + 18}{27} - x$

Step 1.4

Write each expression with a common denominator of $45$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{4 (x + 2)}{15}$ by $\frac{3}{3}$ .

$\frac{4 (x + 2) \cdot 3}{15 \cdot 3} - \frac{2 (x - 4)}{9} \cdot \frac{5}{5} = \frac{27 x + 18}{27} - x$

Step 1.4.2

Multiply $15$ by $3$ .

$\frac{4 (x + 2) \cdot 3}{45} - \frac{2 (x - 4)}{9} \cdot \frac{5}{5} = \frac{27 x + 18}{27} - x$

Step 1.4.3

Multiply $\frac{2 (x - 4)}{9}$ by $\frac{5}{5}$ .

$\frac{4 (x + 2) \cdot 3}{45} - \frac{2 (x - 4) \cdot 5}{9 \cdot 5} = \frac{27 x + 18}{27} - x$

Step 1.4.4

Multiply $9$ by $5$ .

$\frac{4 (x + 2) \cdot 3}{45} - \frac{2 (x - 4) \cdot 5}{45} = \frac{27 x + 18}{27} - x$

Step 1.5

Combine the numerators over the common denominator.

$\frac{4 (x + 2) \cdot 3 - 2 (x - 4) \cdot 5}{45} = \frac{27 x + 18}{27} - x$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Factor $2$ out of $4 (x + 2) \cdot 3 - 2 (x - 4) \cdot 5$ .

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Step 1.6.1.1

Factor $2$ out of $4 (x + 2) \cdot 3$ .

$\frac{2 (2 (x + 2) \cdot 3) - 2 (x - 4) \cdot 5}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.1.2

Factor $2$ out of $- 2 (x - 4) \cdot 5$ .

$\frac{2 (2 (x + 2) \cdot 3) + 2 (- (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.1.3

Factor $2$ out of $2 (2 (x + 2) \cdot 3) + 2 (- (x - 4) \cdot 5)$ .

$\frac{2 (2 (x + 2) \cdot 3 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.2

Apply the distributive property.

$\frac{2 ((2 x + 2 \cdot 2) \cdot 3 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.3

Multiply $2$ by $2$ .

$\frac{2 ((2 x + 4) \cdot 3 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.4

Apply the distributive property.

$\frac{2 (2 x \cdot 3 + 4 \cdot 3 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.5

Multiply $3$ by $2$ .

$\frac{2 (6 x + 4 \cdot 3 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.6

Multiply $4$ by $3$ .

$\frac{2 (6 x + 12 - (x - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.7

Apply the distributive property.

$\frac{2 (6 x + 12 + (- x - - 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.8

Multiply $- 1$ by $- 4$ .

$\frac{2 (6 x + 12 + (- x + 4) \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.9

Apply the distributive property.

$\frac{2 (6 x + 12 - x \cdot 5 + 4 \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.10

Multiply $5$ by $- 1$ .

$\frac{2 (6 x + 12 - 5 x + 4 \cdot 5)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.11

Multiply $4$ by $5$ .

$\frac{2 (6 x + 12 - 5 x + 20)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.12

Subtract $5 x$ from $6 x$ .

$\frac{2 (x + 12 + 20)}{45} = \frac{27 x + 18}{27} - x$

Step 1.6.13

Add $12$ and $20$ .

$\frac{2 (x + 32)}{45} = \frac{27 x + 18}{27} - x$

Step 2

Simplify $\frac{27 x + 18}{27} - x$ .

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Step 2.1

Cancel the common factor of $27 x + 18$ and $27$ .

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Step 2.1.1

Factor $9$ out of $27 x$ .

$\frac{2 (x + 32)}{45} = \frac{9 (3 x) + 18}{27} - x$

Step 2.1.2

Factor $9$ out of $18$ .

$\frac{2 (x + 32)}{45} = \frac{9 (3 x) + 9 (2)}{27} - x$

Step 2.1.3

Factor $9$ out of $9 (3 x) + 9 (2)$ .

$\frac{2 (x + 32)}{45} = \frac{9 (3 x + 2)}{27} - x$

Step 2.1.4

Cancel the common factors.

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Step 2.1.4.1

Factor $9$ out of $27$ .

$\frac{2 (x + 32)}{45} = \frac{9 (3 x + 2)}{9 (3)} - x$

Step 2.1.4.2

Cancel the common factor.

$\frac{2 (x + 32)}{45} = \frac{9 (3 x + 2)}{9 \cdot 3} - x$

Step 2.1.4.3

Rewrite the expression.

$\frac{2 (x + 32)}{45} = \frac{3 x + 2}{3} - x$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 (x + 32)}{45} = \frac{3 x + 2}{3} - x \cdot \frac{3}{3}$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $- x$ and $\frac{3}{3}$ .

$\frac{2 (x + 32)}{45} = \frac{3 x + 2}{3} + \frac{- x \cdot 3}{3}$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{2 (x + 32)}{45} = \frac{3 x + 2 - x \cdot 3}{3}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $3$ by $- 1$ .

$\frac{2 (x + 32)}{45} = \frac{3 x + 2 - 3 x}{3}$

Step 2.4.2

Subtract $3 x$ from $3 x$ .

$\frac{2 (x + 32)}{45} = \frac{0 + 2}{3}$

Step 2.4.3

Add $0$ and $2$ .

$\frac{2 (x + 32)}{45} = \frac{2}{3}$

Step 3

Multiply both sides of the equation by $\frac{45}{2}$ .

$\frac{45}{2} \cdot \frac{2 (x + 32)}{45} = \frac{45}{2} \cdot \frac{2}{3}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{45}{2} \cdot \frac{2 (x + 32)}{45}$ .

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Step 4.1.1.1

Cancel the common factor of $45$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{45}{2} \cdot \frac{2 (x + 32)}{45} = \frac{45}{2} \cdot \frac{2}{3}$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 (x + 32)) = \frac{45}{2} \cdot \frac{2}{3}$

Step 4.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.1.2.1

Cancel the common factor.

$\frac{1}{2} (2 (x + 32)) = \frac{45}{2} \cdot \frac{2}{3}$

Step 4.1.1.2.2

Rewrite the expression.

$x + 32 = \frac{45}{2} \cdot \frac{2}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{45}{2} \cdot \frac{2}{3}$ .

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $45$ .

$x + 32 = \frac{3 (15)}{2} \cdot \frac{2}{3}$

Step 4.2.1.1.2

Cancel the common factor.

$x + 32 = \frac{3 \cdot 15}{2} \cdot \frac{2}{3}$

Step 4.2.1.1.3

Rewrite the expression.

$x + 32 = \frac{15}{2} \cdot 2$

Step 4.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.1

Cancel the common factor.

$x + 32 = \frac{15}{2} \cdot 2$

Step 4.2.1.2.2

Rewrite the expression.

$x + 32 = 15$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Subtract $32$ from both sides of the equation.

$x = 15 - 32$

Step 5.2

Subtract $32$ from $15$ .

$x = - 17$