Question

Solve :

$\frac{2}{x+5}-\frac{1}{x-1}=1$.

Short answer

$x=-1,-2$.

Step-by-step solution

Step 1. Given information.

To solve$\frac{2}{x+5}-\frac{1}{x-1}=1$.

Step 2. Solve by taking LCD.

We get

$\frac{2}{x+5}-\frac{1}{x-1}=1;x\ne -5,x\ne 1$

Taking LCD and multiply it to both sides,

$$\begin{gathered} (x+5)(x-1)[\frac{2}{x+5}-\frac{1}{x-1}]=1\left[\right(x+5\left)\right(x-1\left)\right] \\ 2(x-1)-1(x+5)=(x+5)(x-1) \\ 2x-2-x-5=x^2+5x-x-5 \\ x-7=x^2+4x-5 \\ 0=x^2+3x+2 \end{gathered}$$

Step 3. Factorize and use zero product property.

On simplifying,

$$\begin{gathered} x^2+3x+2=0 \\ {x}^2+2x+x+2=0 \\ x(x+2)+1(x+2)=0 \\ (x+1)(x+2)=0 \\ x+1=0orx+2=0 \\ x=-1orx=-2 \end{gathered}$$