91Ó°ÊÓ

Change the inequality equal to zero to make it easier and then get the value of x
First arrange the inequality,

$\begin{array}{r}\frac{x−1}{x+2}â‰\yen −2\\ \frac{x−1}{x+2}+2â‰\yen 0\\ \frac{x−1}{x+2}+2â‹\dots \frac{x+2}{x+2}â‰\yen 0\\ \frac{x−1+2x+4}{x+2}â‰\yen 0\\ \frac{3x−3}{x+2}â‰\yen 0\end{array}$

Make it equal to zero.

$\begin{array}{r}\frac{3x−3}{x+2}â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ (3x−3)(x+2)â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ 3x−3â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0;x+2â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =0\\ 3xâ€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =3;xâ€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =−2\\ xâ€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =\frac{3}{3}\\ xâ€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots =1\end{array}$

From the obtained values of x, we can form the interval,

So the interval we have is:

$(-∞,-2)∪(-2,1)∪(1,∞)$

Since, $f\left(x\right)â‰\yen 0$, so is positive or zero.

check a number in each interval and evaluate the function to see whether it is satisfying the function.

Since, $12â‰\yen 0,\frac{3}{4}â‰\yen 0$satisfy $f$. Thus,

role="math" localid="1646208744522" $$\begin{gathered} \left\{x\right|x<-2or1≤x≤∞\left\} \\ \right(-∞,-2)∪[1,∞) \end{gathered}$$