Question

Solve the inequality :-

$\frac{ ({2}^{x}-1)(x + 1)(x - 1) }{(x + 2)} \leqslant 0$

Answer 1

Final Answer :
$- 2 < x \leqslant - 1 \: or \: \\ 0\leqslant x \leqslant 1$
Steps and Understanding :
1) Domain of LHS function is :
(x+2)!=0
x! =-2
=> x not equal to -2.

2) Since, the LHS function is trivial so we don't need to simplify it.

Critical Points :
$( {2}^{x} - 1) = 0 \\ = > {2}^{x} = 1 = > x = 0 \\ \\ ( x+ 1) = 0 = > x = - 1 \\ \\ (x - 1) = 0 = > x = 1 \\ \\ (x + 2) = 0 = > x = - 2$

(3) Since, all critical points will occur odd number of times, so there will be alternate change in signs.

For Number line and signs see pic.

(4) If you don't understand signs just put the value between critical points to check it's sign.
Still doubt exist notify me.

(5) We will take those values which contains negative sign and in domain. (see pic)
We got :

$- 2 < x \leqslant - 1 \: or \: \: 0 \leqslant \: x \leqslant 1$
Equality is not in -2 as it is not in domain or
1/(x+2) will not be defined.