Question

Solve $\frac{5}{x-2}>3$ and represent the solution set on the number line.

Or

Solve $|x|<4$ and represent the solution set on the number line.

Answer 1

(i) $\frac{5}{x-2}>3$

$\Rightarrow \frac{5}{x-2}-3>3-3$

$\Rightarrow \frac{5-3x+6}{x-2}>0$

$\Rightarrow \frac{11-3x}{x-2}>0$

Case 1: $11-3x>0$ and $x-2>0$

$\Rightarrow 11>3x$ and $x>2$

$\Rightarrow x<\frac{11}{3}$ and $x>2$

Therefore, solution set is $(2,\frac{11}{3})$

Case 2:$11-3x<0$ and $x-2<0$

$\Rightarrow 11<3x$ and $x<2$

$\Rightarrow x>\frac{11}{3}$ and $x<2$

Therefore, solution set is $\varphi$

Hence, the solution set of the given equation $=(2,\frac{11}{3})\cup \varphi =(2,\frac{11}{3})$

(ii) $|x|<4$

$\Rightarrow -4<x<4$

$\therefore$ Solution set of the given equation is $(-4,4)$