Algebra Examples

$\frac{x - 7}{x^{2} - 16} - \frac{x - 1}{16 - x^{2}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{x - 7}{x^{2} - 4^{2}} - \frac{x - 1}{16 - x^{2}}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{16 - x^{2}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{4^{2} - x^{2}}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(4 + x) (4 - x)}$

Step 2

Simplify with factoring out.

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Step 2.1

Reorder terms.

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (4 - x)}$

Step 2.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (- 1 (- 4) - x)}$

Step 2.3

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

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (- 1 (- 4) - (x))}$

Step 2.4

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (- 1 (- 4 + x))}$

Step 2.5

Reorder terms.

$\frac{x - 7}{(x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (- 1 (x - 4))}$

Step 3

To write $\frac{x - 7}{(x + 4) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{x - 7}{(x + 4) (x - 4)} \cdot \frac{- 1}{- 1} - \frac{x - 1}{(x + 4) (- 1 (x - 4))}$

Step 4

Write each expression with a common denominator of $(x + 4) (x - 4) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x - 7}{(x + 4) (x - 4)}$ by $\frac{- 1}{- 1}$ .

$\frac{(x - 7) \cdot - 1}{(x + 4) (x - 4) \cdot - 1} - \frac{x - 1}{(x + 4) (- 1 (x - 4))}$

Step 4.2

Reorder the factors of $(x + 4) (x - 4) \cdot - 1$ .

$\frac{(x - 7) \cdot - 1}{- (x + 4) (x - 4)} - \frac{x - 1}{(x + 4) (- 1 (x - 4))}$

Step 4.3

Reorder the factors of $(x + 4) (- 1 (x - 4))$ .

$\frac{(x - 7) \cdot - 1}{- (x + 4) (x - 4)} - \frac{x - 1}{- (x + 4) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 7) \cdot - 1 - (x - 1)}{- (x + 4) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot - 1 - 7 \cdot - 1 - (x - 1)}{- (x + 4) (x - 4)}$

Step 6.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x - 7 \cdot - 1 - (x - 1)}{- (x + 4) (x - 4)}$

Step 6.3

Multiply $- 7$ by $- 1$ .

$\frac{- 1 \cdot x + 7 - (x - 1)}{- (x + 4) (x - 4)}$

Step 6.4

Rewrite $- 1 x$ as $- x$ .

$\frac{- x + 7 - (x - 1)}{- (x + 4) (x - 4)}$

Step 6.5

Apply the distributive property.

$\frac{- x + 7 - x - - 1}{- (x + 4) (x - 4)}$

Step 6.6

Multiply $- 1$ by $- 1$ .

$\frac{- x + 7 - x + 1}{- (x + 4) (x - 4)}$

Step 6.7

Subtract $x$ from $- x$ .

$\frac{- 2 x + 7 + 1}{- (x + 4) (x - 4)}$

Step 6.8

Add $7$ and $1$ .

$\frac{- 2 x + 8}{- (x + 4) (x - 4)}$

Step 6.9

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

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Step 6.9.1

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

$\frac{2 (- x) + 8}{- (x + 4) (x - 4)}$

Step 6.9.2

Factor $2$ out of $8$ .

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

Step 6.9.3

Factor $2$ out of $2 (- x) + 2 (4)$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 7.1.3

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 7.1.4

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 7.1.5

Cancel the common factor.

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

Step 7.1.6

Rewrite the expression.

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

Step 7.2

Dividing two negative values results in a positive value.

$\frac{2}{x + 4}$