Simplify (x-1)/(x^2-x-6)-4/(5x+10)

$\frac{x - 1}{x^{2} - x - 6} - \frac{4}{5 x + 10}$

Step 1

Simplify each term.

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$\frac{x - 1}{(x - 3) (x + 2)} - \frac{4}{5 x + 10}$

Factor $5$ out of $5 x + 10$ .

Factor $5$ out of $5 x$ .

$\frac{x - 1}{(x - 3) (x + 2)} - \frac{4}{5 (x) + 10}$

Factor $5$ out of $10$ .

$\frac{x - 1}{(x - 3) (x + 2)} - \frac{4}{5 x + 5 \cdot 2}$

Factor $5$ out of $5 x + 5 \cdot 2$ .

$\frac{x - 1}{(x - 3) (x + 2)} - \frac{4}{5 (x + 2)}$

Step 2

To write $\frac{x - 1}{(x - 3) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x - 1}{(x - 3) (x + 2)} \cdot \frac{5}{5} - \frac{4}{5 (x + 2)}$

Step 3

To write $- \frac{4}{5 (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x - 1}{(x - 3) (x + 2)} \cdot \frac{5}{5} - \frac{4}{5 (x + 2)} \cdot \frac{x - 3}{x - 3}$

Step 4

Write each expression with a common denominator of $(x - 3) (x + 2) \cdot 5$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 1}{(x - 3) (x + 2)}$ by $\frac{5}{5}$ .

$\frac{(x - 1) \cdot 5}{(x - 3) (x + 2) \cdot 5} - \frac{4}{5 (x + 2)} \cdot \frac{x - 3}{x - 3}$

Multiply $\frac{4}{5 (x + 2)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x - 1) \cdot 5}{(x - 3) (x + 2) \cdot 5} - \frac{4 (x - 3)}{5 (x + 2) (x - 3)}$

Reorder the factors of $(x - 3) (x + 2) \cdot 5$ .

$\frac{(x - 1) \cdot 5}{5 (x + 2) (x - 3)} - \frac{4 (x - 3)}{5 (x + 2) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 1) \cdot 5 - 4 (x - 3)}{5 (x + 2) (x - 3)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot 5 - 1 \cdot 5 - 4 (x - 3)}{5 (x + 2) (x - 3)}$

Move $5$ to the left of $x$ .

$\frac{5 \cdot x - 1 \cdot 5 - 4 (x - 3)}{5 (x + 2) (x - 3)}$

Multiply $- 1$ by $5$ .

$\frac{5 \cdot x - 5 - 4 (x - 3)}{5 (x + 2) (x - 3)}$

Apply the distributive property.

$\frac{5 x - 5 - 4 x - 4 \cdot - 3}{5 (x + 2) (x - 3)}$

Multiply $- 4$ by $- 3$ .

$\frac{5 x - 5 - 4 x + 12}{5 (x + 2) (x - 3)}$

Subtract $4 x$ from $5 x$ .

$\frac{x - 5 + 12}{5 (x + 2) (x - 3)}$

Add $- 5$ and $12$ .

$\frac{x + 7}{5 (x + 2) (x - 3)}$

Simplify (x-1)/(x^2-x-6)-4/(5x+10)

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