Question

If $xϵR$, find the solution set of each of the following inequations. Represent it on the number line.
$\frac{x+4}{x-2}>0$

Answer 1

Replace the inequality sign with an equal sign,so that we can solve it like a normal equation.

$\frac{x+4}{x-2}=0$

$\Rightarrow x+4=0$

$\Rightarrow x=-4$

Note that $x$ is undefined at $2$

$\therefore x\ne 2$

From the values of $x$ above, we have these $3$ intervals to test.

$x<-4$

$-4<x<2$ and $x>2$

Let us pick a test point for each interval

For the interval $x<-4$

Let's pick $x=-5$

Then $\frac{-5+4}{-5-2}>0$

$\Rightarrow 0.142857>0$ which is true.

For the interval $-4<x<2$

Let's pick $x=0$

Then $\frac{0+4}{0-2}>0$

$\Rightarrow -2>0$ which is false.

For the interval $x>2$

Let's pick $x=3$

Then $\frac{3+4}{3-2}>0$

$\Rightarrow 7>0$ which is true.

$\therefore x<-4$ and $x>2$

Hence $x\in (-4,2)$