Algebra Examples

$\frac{3}{x - 3} - \frac{5}{x + 2}$

Step 1

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

$\frac{3}{x - 3} \cdot \frac{x + 2}{x + 2} - \frac{5}{x + 2}$

Step 2

To write $- \frac{5}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3}{x - 3} \cdot \frac{x + 2}{x + 2} - \frac{5}{x + 2} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of $(x - 3) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{3 (x + 2)}{(x - 3) (x + 2)} - \frac{5}{x + 2} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply $\frac{5}{x + 2}$ by $\frac{x - 3}{x - 3}$ .

$\frac{3 (x + 2)}{(x - 3) (x + 2)} - \frac{5 (x - 3)}{(x + 2) (x - 3)}$

Step 3.3

Reorder the factors of $(x - 3) (x + 2)$ .

$\frac{3 (x + 2)}{(x + 2) (x - 3)} - \frac{5 (x - 3)}{(x + 2) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (x + 2) - 5 (x - 3)}{(x + 2) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 2 - 5 (x - 3)}{(x + 2) (x - 3)}$

Step 5.2

Multiply $3$ by $2$ .

$\frac{3 x + 6 - 5 (x - 3)}{(x + 2) (x - 3)}$

Step 5.3

Apply the distributive property.

$\frac{3 x + 6 - 5 x - 5 \cdot - 3}{(x + 2) (x - 3)}$

Step 5.4

Multiply $- 5$ by $- 3$ .

$\frac{3 x + 6 - 5 x + 15}{(x + 2) (x - 3)}$

Step 5.5

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x + 6 + 15}{(x + 2) (x - 3)}$

Step 5.6

Add $6$ and $15$ .

$\frac{- 2 x + 21}{(x + 2) (x - 3)}$

Step 6

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

$\frac{- (2 x) + 21}{(x + 2) (x - 3)}$

Step 7

Rewrite $21$ as $- 1 (- 21)$ .

$\frac{- (2 x) - 1 (- 21)}{(x + 2) (x - 3)}$

Step 8

Factor $- 1$ out of $- (2 x) - 1 (- 21)$ .

$\frac{- (2 x - 21)}{(x + 2) (x - 3)}$

Step 9

Rewrite $- (2 x - 21)$ as $- 1 (2 x - 21)$ .

$\frac{- 1 (2 x - 21)}{(x + 2) (x - 3)}$

Step 10

Move the negative in front of the fraction.

$- \frac{2 x - 21}{(x + 2) (x - 3)}$