Solve by Factoring 1/(x+2)-1/(x-1)=0

$\frac{1}{x + 2} - \frac{1}{x - 1} = 0$

Step 1

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

$\frac{1}{x + 2} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} = 0$

Step 2

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

$\frac{1}{x + 2} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{x + 2}{x + 2} = 0$

Step 3

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

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

$\frac{x - 1}{(x + 2) (x - 1)} - \frac{1}{x - 1} \cdot \frac{x + 2}{x + 2} = 0$

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

$\frac{x - 1}{(x + 2) (x - 1)} - \frac{x + 2}{(x - 1) (x + 2)} = 0$

Reorder the factors of $(x - 1) (x + 2)$ .

$\frac{x - 1}{(x + 2) (x - 1)} - \frac{x + 2}{(x + 2) (x - 1)} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x - 1 - (x + 2)}{(x + 2) (x - 1)} = 0$

Step 5

Rewrite $\frac{x - 1 - (x + 2)}{(x + 2) (x - 1)}$ in a factored form.

Apply the distributive property.

$\frac{x - 1 - x - 1 \cdot 2}{(x + 2) (x - 1)} = 0$

Multiply $- 1$ by $2$ .

$\frac{x - 1 - x - 2}{(x + 2) (x - 1)} = 0$

Subtract $x$ from $x$ .

$\frac{0 - 1 - 2}{(x + 2) (x - 1)} = 0$

Subtract $1$ from $0$ .

$\frac{- 1 - 2}{(x + 2) (x - 1)} = 0$

Subtract $2$ from $- 1$ .

$\frac{- 3}{(x + 2) (x - 1)} = 0$

Step 6

Move the negative in front of the fraction.

$- \frac{3}{(x + 2) (x - 1)} = 0$

Step 7

Set the numerator equal to zero.

$3 = 0$

Step 8

Since $3 \neq 0$ , there are no solutions.

No solution