Simplify 5/(x-1)-(7x)/(x^2+2x-3)

$\frac{5}{x - 1} - \frac{7 x}{x^{2} + 2 x - 3}$

Step 1

Factor $x^{2} + 2 x - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

$\frac{5}{x - 1} - \frac{7 x}{(x - 1) (x + 3)}$

Step 2

To write $\frac{5}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{5}{x - 1} \cdot \frac{x + 3}{x + 3} - \frac{7 x}{(x - 1) (x + 3)}$

Step 3

Simplify terms.

Multiply $\frac{5}{x - 1}$ by $\frac{x + 3}{x + 3}$ .

$\frac{5 (x + 3)}{(x - 1) (x + 3)} - \frac{7 x}{(x - 1) (x + 3)}$

Combine the numerators over the common denominator.

$\frac{5 (x + 3) - 7 x}{(x - 1) (x + 3)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{5 x + 5 \cdot 3 - 7 x}{(x - 1) (x + 3)}$

Multiply $5$ by $3$ .

$\frac{5 x + 15 - 7 x}{(x - 1) (x + 3)}$

Subtract $7 x$ from $5 x$ .

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

Step 5

Simplify with factoring out.

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

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

Rewrite $15$ as $- 1 (- 15)$ .

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

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

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

Simplify the expression.

Rewrite $- (2 x - 15)$ as $- 1 (2 x - 15)$ .

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

Move the negative in front of the fraction.

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

Simplify 5/(x-1)-(7x)/(x^2+2x-3)

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