Question

Solve $\frac{x}{x-9}\le \frac{1}{1-x}$
Write your solution set in interval notation.

• A. $(-3<x<1)\cap (3<x<9)$
• B. $(-3\le x<1)\cap (3<x<9)$
• C. $(-3\le x<1)\cap (3<x<9)$
• D. $(-3\le x\le 1)\cap (3\le x\le 9)$

Answer 1

The correct option is A $(-3<x<1)\cap (3<x<9)$

$\text{solution:}\frac{x}{x-9}⩽\frac{1}{1-x}$

$\Rightarrow \frac{x}{x-9}-\frac{1}{1-x}⩽0$

$\Rightarrow \frac{x-x^2-x+9}{(x-9)(1-x)}⩽0$

$\frac{(9-x^2)}{(x-9)(1-x)}⩽0$

$\Rightarrow \frac{(x^2-9)}{(x-9)(x-1)}⩽0$

$\Rightarrow \frac{(x-3)(x+3)}{(x-9)(x-1)}⩽0$

$x\in \left[\begin{array}{cc}-3& 1\end{array}\right)\cup \left[\begin{array}{c}39\end{array}\right)$

or $(-3⩽x<1)\cup (3⩽x<9)$