Question

Solve the following inequality:
$\frac{x-1}{x}-\frac{x+1}{x-1}<2$

• A. (0,0) ∪ (½,1)
• B. (-1,0) ∪ (½,2)
• C. (-2,0) ∪ (½,1)
• D. (-1,0) ∪ (½,1)

Answer 1

The correct option is D (-1,0) ∪ (½,1)

We have,
$\frac{x-1}{x}-\frac{x+1}{x-1}<2$

$\Rightarrow \frac{x-1}{x}-\frac{x+1}{x-1}-2<0$

$\Rightarrow \frac{(x-1{)}^2-x(x+1)-2x(x-1)}{x(x-1)}<0$

$\Rightarrow \frac{-2x^2-x+1}{x(x-1)}<0$

$\Rightarrow \frac{-2x^2-2x+x+1}{x(x-1)}>0$

$\Rightarrow \frac{(x+1)(-2x+1)}{x(x-1)}>0$ $[x\ne 0,x\ne 1]$

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

$xϵ(-1,0)\cup (\frac{1}{2},1)$


So, option d is correct.