Precalculus Examples

$\frac{3}{x - 6} + \frac{4}{x - 4}$

Step 1

To write $\frac{3}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

Step 2

To write $\frac{4}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

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

Step 3

Write each expression with a common denominator of $(x - 6) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3}{x - 6}$ by $\frac{x - 4}{x - 4}$ .

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

Step 3.2

Multiply $\frac{4}{x - 4}$ by $\frac{x - 6}{x - 6}$ .

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

Step 3.3

Reorder the factors of $(x - 6) (x - 4)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $3$ by $- 4$ .

$\frac{3 x - 12 + 4 (x - 6)}{(x - 4) (x - 6)}$

Step 5.3

Apply the distributive property.

$\frac{3 x - 12 + 4 x + 4 \cdot - 6}{(x - 4) (x - 6)}$

Step 5.4

Multiply $4$ by $- 6$ .

$\frac{3 x - 12 + 4 x - 24}{(x - 4) (x - 6)}$

Step 5.5

Add $3 x$ and $4 x$ .

$\frac{7 x - 12 - 24}{(x - 4) (x - 6)}$

Step 5.6

Subtract $24$ from $- 12$ .

$\frac{7 x - 36}{(x - 4) (x - 6)}$