Solve the following system of equations in R. |x+2|-x x <2 - Vidyadip

Question

Solve the following system of equations in R.
$\frac{|\text{x}+2|-\text{x}}{\text{x}}<2$

✓

Answer

We have,
$\frac{|\text{x}+2|-\text{x}}{\text{x}}<0$
$\frac{|\text{x}+2|-\text{x}}{\text{x}}-2<0$
$\frac{|\text{x}+2|-\text{x}-2\text{x}}{\text{x}}<0$
$\frac{|\text{x}+2|-3\text{x}}{\text{x}}<0\ ...(\text{i})$
Case 1: When $|\text{x}+2|\geq0$
$\Rightarrow\frac{\text{x}+2-3\text{x}}{\text{x}}<0$
⇒ - 2x + 2 < 0
⇒ -2x < - 2 and x > 0
⇒ x > 1 ...(ii)
case2: |x + 2|< 0
x < -2
⇒ -(x + 2)- 3x < 0
⇒ - x - 2 - 3x < 0
⇒ -4x - 2 < 0
⇒ -4x < 2
$\Rightarrow\text{x}>\frac{-1}{2}\ ...(\text{iii})$
and x < 0
Combining (ii) and (iii), we get $(-\infty,0)\cup(1,\infty)$ as the solution set.

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