Pre-Algebra Examples

$\frac{5}{x + 5} + \frac{3}{x - 5}$

Step 1

To write $\frac{5}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{5}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{3}{x - 5}$

Step 2

To write $\frac{3}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{5}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{3}{x - 5} \cdot \frac{x + 5}{x + 5}$

Step 3

Write each expression with a common denominator of $(x + 5) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

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

Step 3.2

Multiply $\frac{3}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

$\frac{5 (x - 5)}{(x + 5) (x - 5)} + \frac{3 (x + 5)}{(x - 5) (x + 5)}$

Step 3.3

Reorder the factors of $(x - 5) (x + 5)$ .

$\frac{5 (x - 5)}{(x + 5) (x - 5)} + \frac{3 (x + 5)}{(x + 5) (x - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 (x - 5) + 3 (x + 5)}{(x + 5) (x - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $5$ by $- 5$ .

$\frac{5 x - 25 + 3 (x + 5)}{(x + 5) (x - 5)}$

Step 5.3

Apply the distributive property.

$\frac{5 x - 25 + 3 x + 3 \cdot 5}{(x + 5) (x - 5)}$

Step 5.4

Multiply $3$ by $5$ .

$\frac{5 x - 25 + 3 x + 15}{(x + 5) (x - 5)}$

Step 5.5

Add $5 x$ and $3 x$ .

$\frac{8 x - 25 + 15}{(x + 5) (x - 5)}$

Step 5.6

Add $- 25$ and $15$ .

$\frac{8 x - 10}{(x + 5) (x - 5)}$

Step 5.7

Factor $2$ out of $8 x - 10$ .

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Step 5.7.1

Factor $2$ out of $8 x$ .

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

Step 5.7.2

Factor $2$ out of $- 10$ .

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

Step 5.7.3

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

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