Algebra Examples

$\frac{\frac{5}{x + 6} - \frac{2 x}{2 x - 1}}{\frac{x}{2 x - 1} + \frac{4}{x + 6}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 6) (2 x - 1)$ .

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Step 1.1

Multiply $\frac{\frac{5}{x + 6} - \frac{2 x}{2 x - 1}}{\frac{x}{2 x - 1} + \frac{4}{x + 6}}$ by $\frac{(x + 6) (2 x - 1)}{(x + 6) (2 x - 1)}$ .

$\frac{(x + 6) (2 x - 1)}{(x + 6) (2 x - 1)} \cdot \frac{\frac{5}{x + 6} - \frac{2 x}{2 x - 1}}{\frac{x}{2 x - 1} + \frac{4}{x + 6}}$

Step 1.2

Combine.

$\frac{(x + 6) (2 x - 1) (\frac{5}{x + 6} - \frac{2 x}{2 x - 1})}{(x + 6) (2 x - 1) (\frac{x}{2 x - 1} + \frac{4}{x + 6})}$

Step 2

Apply the distributive property.

$\frac{(x + 6) (2 x - 1) \frac{5}{x + 6} + (x + 6) (2 x - 1) (- \frac{2 x}{2 x - 1})}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 6$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x + 6) (2 x - 1) \frac{5}{x + 6} + (x + 6) (2 x - 1) (- \frac{2 x}{2 x - 1})}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.1.2

Rewrite the expression.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (2 x - 1) (- \frac{2 x}{2 x - 1})}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.2

Cancel the common factor of $2 x - 1$ .

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Step 3.2.1

Move the leading negative in $- \frac{2 x}{2 x - 1}$ into the numerator.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (2 x - 1) \frac{- 2 x}{2 x - 1}}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.2.2

Factor $2 x - 1$ out of $(x + 6) (2 x - 1)$ .

$\frac{(2 x - 1) \cdot 5 + (2 x - 1) (x + 6) \frac{- 2 x}{2 x - 1}}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.2.3

Cancel the common factor.

$\frac{(2 x - 1) \cdot 5 + (2 x - 1) (x + 6) \frac{- 2 x}{2 x - 1}}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.2.4

Rewrite the expression.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(x + 6) (2 x - 1) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.3

Cancel the common factor of $2 x - 1$ .

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Step 3.3.1

Factor $2 x - 1$ out of $(x + 6) (2 x - 1)$ .

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(2 x - 1) (x + 6) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.3.2

Cancel the common factor.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(2 x - 1) (x + 6) \frac{x}{2 x - 1} + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.3.3

Rewrite the expression.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(x + 6) x + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.4

Cancel the common factor of $x + 6$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(x + 6) x + (x + 6) (2 x - 1) \frac{4}{x + 6}}$

Step 3.4.2

Rewrite the expression.

$\frac{(2 x - 1) \cdot 5 + (x + 6) (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{2 x \cdot 5 - 1 \cdot 5 + (x + 6) (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.2

Multiply $5$ by $2$ .

$\frac{10 x - 1 \cdot 5 + (x + 6) (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.3

Multiply $- 1$ by $5$ .

$\frac{10 x - 5 + (x + 6) (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.4

Apply the distributive property.

$\frac{10 x - 5 + x (- 2 x) + 6 (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.5

Rewrite using the commutative property of multiplication.

$\frac{10 x - 5 - 2 x \cdot x + 6 (- 2 x)}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.6

Multiply $- 2$ by $6$ .

$\frac{10 x - 5 - 2 x \cdot x - 12 x}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.7

Multiply $x$ by $x$ by adding the exponents.

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Step 4.7.1

Move $x$ .

$\frac{10 x - 5 - 2 (x \cdot x) - 12 x}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.7.2

Multiply $x$ by $x$ .

$\frac{10 x - 5 - 2 x^{2} - 12 x}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.8

Subtract $12 x$ from $10 x$ .

$\frac{- 2 x - 5 - 2 x^{2}}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 4.9

Reorder terms.

$\frac{- 2 x^{2} - 2 x - 5}{(x + 6) x + (2 x - 1) \cdot 4}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{- 2 x^{2} - 2 x - 5}{x \cdot x + 6 x + (2 x - 1) \cdot 4}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{- 2 x^{2} - 2 x - 5}{x^{2} + 6 x + (2 x - 1) \cdot 4}$

Step 5.3

Apply the distributive property.

$\frac{- 2 x^{2} - 2 x - 5}{x^{2} + 6 x + 2 x \cdot 4 - 1 \cdot 4}$

Step 5.4

Multiply $4$ by $2$ .

$\frac{- 2 x^{2} - 2 x - 5}{x^{2} + 6 x + 8 x - 1 \cdot 4}$

Step 5.5

Multiply $- 1$ by $4$ .

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

Step 5.6

Add $6 x$ and $8 x$ .

$\frac{- 2 x^{2} - 2 x - 5}{x^{2} + 14 x - 4}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{- (2 x^{2}) - 2 x - 5}{x^{2} + 14 x - 4}$

Step 6.2

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

$\frac{- (2 x^{2}) - (2 x) - 5}{x^{2} + 14 x - 4}$

Step 6.3

Factor $- 1$ out of $- (2 x^{2}) - (2 x)$ .

$\frac{- (2 x^{2} + 2 x) - 5}{x^{2} + 14 x - 4}$

Step 6.4

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (2 x^{2} + 2 x) - 1 (5)}{x^{2} + 14 x - 4}$

Step 6.5

Factor $- 1$ out of $- (2 x^{2} + 2 x) - 1 (5)$ .

$\frac{- (2 x^{2} + 2 x + 5)}{x^{2} + 14 x - 4}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (2 x^{2} + 2 x + 5)$ as $- 1 (2 x^{2} + 2 x + 5)$ .

$\frac{- 1 (2 x^{2} + 2 x + 5)}{x^{2} + 14 x - 4}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{2 x^{2} + 2 x + 5}{x^{2} + 14 x - 4}$