Algebra Examples

$(\frac{1}{3} x + \frac{1}{4}) - (\frac{1}{2} x - \frac{2}{3})$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + \frac{1}{4} - (\frac{1}{2} x - \frac{2}{3})$

Step 1.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{3} + \frac{1}{4} - (\frac{x}{2} - \frac{2}{3})$

Step 1.3

Apply the distributive property.

$\frac{x}{3} + \frac{1}{4} - \frac{x}{2} - - \frac{2}{3}$

Step 1.4

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

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{x}{3} + \frac{1}{4} - \frac{x}{2} + 1 (\frac{2}{3})$

Step 1.4.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{x}{3} + \frac{1}{4} - \frac{x}{2} + \frac{2}{3}$

Step 2

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

$\frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} + \frac{1}{4} + \frac{2}{3}$

Step 3

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

$\frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{4} + \frac{2}{3}$

Step 4

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

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Step 4.1

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

$\frac{x \cdot 2}{3 \cdot 2} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{4} + \frac{2}{3}$

Step 4.2

Multiply $3$ by $2$ .

$\frac{x \cdot 2}{6} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{4} + \frac{2}{3}$

Step 4.3

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

$\frac{x \cdot 2}{6} - \frac{x \cdot 3}{2 \cdot 3} + \frac{1}{4} + \frac{2}{3}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{x \cdot 2}{6} - \frac{x \cdot 3}{6} + \frac{1}{4} + \frac{2}{3}$

Step 5

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x \cdot 3}{6} + \frac{1}{4} + \frac{2}{3}$

Step 6

Find the common denominator.

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Step 6.1

Multiply $\frac{x \cdot 2 - x \cdot 3}{6}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2 - x \cdot 3}{6} \cdot \frac{2}{2} + \frac{1}{4} + \frac{2}{3}$

Step 6.2

Multiply $\frac{x \cdot 2 - x \cdot 3}{6}$ by $\frac{2}{2}$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{6 \cdot 2} + \frac{1}{4} + \frac{2}{3}$

Step 6.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{6 \cdot 2} + \frac{1}{4} \cdot \frac{3}{3} + \frac{2}{3}$

Step 6.4

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

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

Step 6.5

Multiply $\frac{2}{3}$ by $\frac{4}{4}$ .

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

Step 6.6

Multiply $\frac{2}{3}$ by $\frac{4}{4}$ .

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

Step 6.7

Reorder the factors of $6 \cdot 2$ .

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

Step 6.8

Multiply $2$ by $6$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{12} + \frac{3}{4 \cdot 3} + \frac{2 \cdot 4}{3 \cdot 4}$

Step 6.9

Reorder the factors of $4 \cdot 3$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{12} + \frac{3}{3 \cdot 4} + \frac{2 \cdot 4}{3 \cdot 4}$

Step 6.10

Multiply $3$ by $4$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{12} + \frac{3}{12} + \frac{2 \cdot 4}{3 \cdot 4}$

Step 6.11

Multiply $3$ by $4$ .

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2}{12} + \frac{3}{12} + \frac{2 \cdot 4}{12}$

Step 7

Combine the numerators over the common denominator.

$\frac{(x \cdot 2 - x \cdot 3) \cdot 2 + 3 + 2 \cdot 4}{12}$

Step 8

Simplify each term.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

$\frac{(2 \cdot x - x \cdot 3) \cdot 2 + 3 + 2 \cdot 4}{12}$

Step 8.1.2

Multiply $3$ by $- 1$ .

$\frac{(2 x - 3 x) \cdot 2 + 3 + 2 \cdot 4}{12}$

Step 8.2

Subtract $3 x$ from $2 x$ .

$\frac{- x \cdot 2 + 3 + 2 \cdot 4}{12}$

Step 8.3

Multiply $2$ by $- 1$ .

$\frac{- 2 x + 3 + 2 \cdot 4}{12}$

Step 8.4

Multiply $2$ by $4$ .

$\frac{- 2 x + 3 + 8}{12}$

Step 9

Simplify with factoring out.

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Step 9.1

Add $3$ and $8$ .

$\frac{- 2 x + 11}{12}$

Step 9.2

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) + 11}{12}$

Step 9.3

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{- (2 x) - 1 (- 11)}{12}$

Step 9.4

Factor $- 1$ out of $- (2 x) - 1 (- 11)$ .

$\frac{- (2 x - 11)}{12}$

Step 9.5

Simplify the expression.

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Step 9.5.1

Rewrite $- (2 x - 11)$ as $- 1 (2 x - 11)$ .

$\frac{- 1 (2 x - 11)}{12}$

Step 9.5.2

Move the negative in front of the fraction.

$- \frac{2 x - 11}{12}$