Question

Solve the following system of inequality:
$|x^2+5x|<6,|x+1|⩽1$

Answer 1

$\mid x^2+5x\mid <6$

$\Rightarrow -6<x^2+5x<6$

a. $-6<x^2+5x$

$\Rightarrow x^2+5x+6>0$

$\Rightarrow (x+3)(x+2)>0$

$\Rightarrow x\in (-\infty ,2)\cup (3,\infty )$ -Equation a

b. $x^2+5x<6$

$\Rightarrow x^2+5x-6<0$

$\Rightarrow (x+6)(x-1)<0$

$\Rightarrow x\in (-6,1)$ - Equation b

For 1, intersection of a & b is taken, $x\in (-6,1)$ - Equation 1.

2. $\mid x+1\mid \le 1$

$\Rightarrow -1\le x+1\le 1$

$\Rightarrow -2\le x\le 0$

$\Rightarrow x\in [-2,0]$ - Equation2.

For the complete inequality, intersection of 1 & 2 is taken,

$\Rightarrow x\in [-2,0]$