Simplify (2x)/(x-7)-x/(x-2)

$\frac{2 x}{x - 7} - \frac{x}{x - 2}$

Step 1

To write $\frac{2 x}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{2 x}{x - 7} \cdot \frac{x - 2}{x - 2} - \frac{x}{x - 2}$

Step 2

To write $- \frac{x}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{2 x}{x - 7} \cdot \frac{x - 2}{x - 2} - \frac{x}{x - 2} \cdot \frac{x - 7}{x - 7}$

Step 3

Write each expression with a common denominator of $(x - 7) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x}{x - 7}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 x (x - 2)}{(x - 7) (x - 2)} - \frac{x}{x - 2} \cdot \frac{x - 7}{x - 7}$

Multiply $\frac{x}{x - 2}$ by $\frac{x - 7}{x - 7}$ .

$\frac{2 x (x - 2)}{(x - 7) (x - 2)} - \frac{x (x - 7)}{(x - 2) (x - 7)}$

Reorder the factors of $(x - 7) (x - 2)$ .

$\frac{2 x (x - 2)}{(x - 2) (x - 7)} - \frac{x (x - 7)}{(x - 2) (x - 7)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 x (x - 2) - x (x - 7)}{(x - 2) (x - 7)}$

Step 5

Simplify the numerator.

Factor $x$ out of $2 x (x - 2) - x (x - 7)$ .

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

$\frac{x (2 (x - 2)) - x (x - 7)}{(x - 2) (x - 7)}$

Factor $x$ out of $- x (x - 7)$ .

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

Factor $x$ out of $x (2 (x - 2)) + x (- (x - 7))$ .

$\frac{x (2 (x - 2) - (x - 7))}{(x - 2) (x - 7)}$

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

$\frac{x (2 x - 4 - (x - 7))}{(x - 2) (x - 7)}$

Apply the distributive property.

$\frac{x (2 x - 4 - x - - 7)}{(x - 2) (x - 7)}$

Multiply $- 1$ by $- 7$ .

$\frac{x (2 x - 4 - x + 7)}{(x - 2) (x - 7)}$

Subtract $x$ from $2 x$ .

$\frac{x (x - 4 + 7)}{(x - 2) (x - 7)}$

Add $- 4$ and $7$ .

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