Question

The values of $x$ for which $\left|\frac{2}{x-4}\right|>1$ holds true is

• A. $(2,4)$
• B. $(2,6)$
• C. $(4,6)$
• D. $(2,4)\cup (4,6)$

Answer 1

The correct option is D $(2,4)\cup (4,6)$

Clearly, $x\ne 4$
When $\frac{2}{x-4}>0$ i.e., $x>4$
$\frac{2}{x-4}>1$
$\Rightarrow \frac{x-4}{2}<1$
$\Rightarrow x-4<2$
$\Rightarrow x<6$
$\Rightarrow x\in (4,6)\cdots \left(1\right)$

When $\frac{2}{x-4}<0$ i.e., $x<4$
$\frac{-2}{x-4}>1$
$\Rightarrow \frac{x-4}{-2}<1$
$\Rightarrow x-4>-2$
$\Rightarrow x>2$
$\Rightarrow x\in (2,4)\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$x\in (2,6)–\left\{4\right\}$