Finite Math Examples

$\frac{9}{x - 3} - \frac{2 x}{x + 1}$

Step 1

To write $\frac{9}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{9}{x - 3} \cdot \frac{x + 1}{x + 1} - \frac{2 x}{x + 1}$

Step 2

To write $- \frac{2 x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{9}{x - 3} \cdot \frac{x + 1}{x + 1} - \frac{2 x}{x + 1} \cdot \frac{x - 3}{x - 3}$

Step 3

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

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Step 3.1

Multiply $\frac{9}{x - 3}$ by $\frac{x + 1}{x + 1}$ .

$\frac{9 (x + 1)}{(x - 3) (x + 1)} - \frac{2 x}{x + 1} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply $\frac{2 x}{x + 1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{9 (x + 1)}{(x - 3) (x + 1)} - \frac{2 x (x - 3)}{(x + 1) (x - 3)}$

Step 3.3

Reorder the factors of $(x - 3) (x + 1)$ .

$\frac{9 (x + 1)}{(x + 1) (x - 3)} - \frac{2 x (x - 3)}{(x + 1) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{9 (x + 1) - 2 x (x - 3)}{(x + 1) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{9 x + 9 \cdot 1 - 2 x (x - 3)}{(x + 1) (x - 3)}$

Step 5.2

Multiply $9$ by $1$ .

$\frac{9 x + 9 - 2 x (x - 3)}{(x + 1) (x - 3)}$

Step 5.3

Apply the distributive property.

$\frac{9 x + 9 - 2 x \cdot x - 2 x \cdot - 3}{(x + 1) (x - 3)}$

Step 5.4

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

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Step 5.4.1

Move $x$ .

$\frac{9 x + 9 - 2 (x \cdot x) - 2 x \cdot - 3}{(x + 1) (x - 3)}$

Step 5.4.2

Multiply $x$ by $x$ .

$\frac{9 x + 9 - 2 x^{2} - 2 x \cdot - 3}{(x + 1) (x - 3)}$

Step 5.5

Multiply $- 3$ by $- 2$ .

$\frac{9 x + 9 - 2 x^{2} + 6 x}{(x + 1) (x - 3)}$

Step 5.6

Add $9 x$ and $6 x$ .

$\frac{15 x + 9 - 2 x^{2}}{(x + 1) (x - 3)}$

Step 5.7

Reorder terms.

$\frac{- 2 x^{2} + 15 x + 9}{(x + 1) (x - 3)}$

Step 6

Simplify with factoring out.

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Step 6.1

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

$\frac{- (2 x^{2}) + 15 x + 9}{(x + 1) (x - 3)}$

Step 6.2

Factor $- 1$ out of $15 x$ .

$\frac{- (2 x^{2}) - (- 15 x) + 9}{(x + 1) (x - 3)}$

Step 6.3

Factor $- 1$ out of $- (2 x^{2}) - (- 15 x)$ .

$\frac{- (2 x^{2} - 15 x) + 9}{(x + 1) (x - 3)}$

Step 6.4

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (2 x^{2} - 15 x) - 1 (- 9)}{(x + 1) (x - 3)}$

Step 6.5

Factor $- 1$ out of $- (2 x^{2} - 15 x) - 1 (- 9)$ .

$\frac{- (2 x^{2} - 15 x - 9)}{(x + 1) (x - 3)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (2 x^{2} - 15 x - 9)$ as $- 1 (2 x^{2} - 15 x - 9)$ .

$\frac{- 1 (2 x^{2} - 15 x - 9)}{(x + 1) (x - 3)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{2 x^{2} - 15 x - 9}{(x + 1) (x - 3)}$