Question

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

Answer 1

$\frac{5}{x-2}>3\Rightarrow \frac{5}{x-2}-3>0$ [adding -3 to both sides]

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

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

$\therefore$ either $(11-3x>0andx-2>0)or(11-3x<0andx-2<0)$

When 11-3x>0 and x-2>0

Now, 11-3x>0 and x-2>0

$\Rightarrow -3x>-11andx>2$

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

$\Rightarrow 2<x<\frac{11}{3}\dots \left(i\right)$

Case II

When 11-3x<0 and x-2<0

Now, 11-3x<0 and x-2<0

$\Rightarrow -3x<-11andx<2$

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

This is not possible, as we cannot find a real number which greater than $\frac{11}{3}$ and less than 2.

$\therefore$ Solution set =$\{xϵR:2<x<\frac{11}{3}\}=(2,\frac{11}{3})$

We can represent this set on the number line, as given below.