Algebra Examples

$\frac{x}{3} + \frac{4 x + 8}{8} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1

Simplify $\frac{x}{3} + \frac{4 x + 8}{8}$ .

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Step 1.1

Cancel the common factor of $4 x + 8$ and $8$ .

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Step 1.1.1

Factor $4$ out of $4 x$ .

$\frac{x}{3} + \frac{4 (x) + 8}{8} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.1.2

Factor $4$ out of $8$ .

$\frac{x}{3} + \frac{4 x + 4 \cdot 2}{8} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.1.3

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

$\frac{x}{3} + \frac{4 (x + 2)}{8} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.1.4

Cancel the common factors.

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Step 1.1.4.1

Factor $4$ out of $8$ .

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

Step 1.1.4.2

Cancel the common factor.

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

Step 1.1.4.3

Rewrite the expression.

$\frac{x}{3} + \frac{x + 2}{2} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.2

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

$\frac{x}{3} \cdot \frac{2}{2} + \frac{x + 2}{2} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.3

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

$\frac{x}{3} \cdot \frac{2}{2} + \frac{x + 2}{2} \cdot \frac{3}{3} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{3 \cdot 2} + \frac{x + 2}{2} \cdot \frac{3}{3} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{x \cdot 2}{6} + \frac{x + 2}{2} \cdot \frac{3}{3} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.4.3

Multiply $\frac{x + 2}{2}$ by $\frac{3}{3}$ .

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

Step 1.4.4

Multiply $2$ by $3$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Move $2$ to the left of $x$ .

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

Step 1.6.2

Apply the distributive property.

$\frac{2 x + x \cdot 3 + 2 \cdot 3}{6} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.6.3

Move $3$ to the left of $x$ .

$\frac{2 x + 3 \cdot x + 2 \cdot 3}{6} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.6.4

Multiply $2$ by $3$ .

$\frac{2 x + 3 \cdot x + 6}{6} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 1.6.5

Add $2 x$ and $3 x$ .

$\frac{5 x + 6}{6} = \frac{x - 5}{6} - \frac{3 - 2 x}{2}$

Step 2

Simplify $\frac{x - 5}{6} - \frac{3 - 2 x}{2}$ .

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Step 2.1

To write $- \frac{3 - 2 x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x + 6}{6} = \frac{x - 5}{6} - \frac{3 - 2 x}{2} \cdot \frac{3}{3}$

Step 2.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{3 - 2 x}{2}$ by $\frac{3}{3}$ .

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

Step 2.2.2

Multiply $2$ by $3$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{5 x + 6}{6} = \frac{x - 5 + (- 1 \cdot 3 - (- 2 x)) \cdot 3}{6}$

Step 2.4.2

Multiply $- 1$ by $3$ .

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

Step 2.4.3

Multiply $- 2$ by $- 1$ .

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

Step 2.4.4

Apply the distributive property.

$\frac{5 x + 6}{6} = \frac{x - 5 - 3 \cdot 3 + 2 x \cdot 3}{6}$

Step 2.4.5

Multiply $- 3$ by $3$ .

$\frac{5 x + 6}{6} = \frac{x - 5 - 9 + 2 x \cdot 3}{6}$

Step 2.4.6

Multiply $3$ by $2$ .

$\frac{5 x + 6}{6} = \frac{x - 5 - 9 + 6 x}{6}$

Step 2.4.7

Add $x$ and $6 x$ .

$\frac{5 x + 6}{6} = \frac{7 x - 5 - 9}{6}$

Step 2.4.8

Subtract $9$ from $- 5$ .

$\frac{5 x + 6}{6} = \frac{7 x - 14}{6}$

Step 2.4.9

Factor $7$ out of $7 x - 14$ .

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Step 2.4.9.1

Factor $7$ out of $7 x$ .

$\frac{5 x + 6}{6} = \frac{7 (x) - 14}{6}$

Step 2.4.9.2

Factor $7$ out of $- 14$ .

$\frac{5 x + 6}{6} = \frac{7 x + 7 \cdot - 2}{6}$

Step 2.4.9.3

Factor $7$ out of $7 x + 7 \cdot - 2$ .

$\frac{5 x + 6}{6} = \frac{7 (x - 2)}{6}$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $\frac{7 (x - 2)}{6}$ from both sides of the equation.

$\frac{5 x + 6}{6} - \frac{7 (x - 2)}{6} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 x + 6 - 7 (x - 2)}{6} = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{5 x + 6 - 7 x - 7 \cdot - 2}{6} = 0$

Step 3.3.2

Multiply $- 7$ by $- 2$ .

$\frac{5 x + 6 - 7 x + 14}{6} = 0$

Step 3.4

Subtract $7 x$ from $5 x$ .

$\frac{- 2 x + 6 + 14}{6} = 0$

Step 3.5

Add $6$ and $14$ .

$\frac{- 2 x + 20}{6} = 0$

Step 3.6

Cancel the common factor of $- 2 x + 20$ and $6$ .

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Step 3.6.1

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

$\frac{2 (- x) + 20}{6} = 0$

Step 3.6.2

Factor $2$ out of $20$ .

$\frac{2 (- x) + 2 (10)}{6} = 0$

Step 3.6.3

Factor $2$ out of $2 (- x) + 2 (10)$ .

$\frac{2 (- x + 10)}{6} = 0$

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $2$ out of $6$ .

$\frac{2 (- x + 10)}{2 (3)} = 0$

Step 3.6.4.2

Cancel the common factor.

$\frac{2 (- x + 10)}{2 \cdot 3} = 0$

Step 3.6.4.3

Rewrite the expression.

$\frac{- x + 10}{3} = 0$

Step 3.7

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 10}{3} = 0$

Step 3.8

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (x) - 1 (- 10)}{3} = 0$

Step 3.9

Factor $- 1$ out of $- (x) - 1 (- 10)$ .

$\frac{- (x - 10)}{3} = 0$

Step 3.10

Rewrite $- (x - 10)$ as $- 1 (x - 10)$ .

$\frac{- 1 (x - 10)}{3} = 0$

Step 3.11

Move the negative in front of the fraction.

$- \frac{x - 10}{3} = 0$

Step 4

Set the numerator equal to zero.

$x - 10 = 0$

Step 5

Add $10$ to both sides of the equation.

$x = 10$