Question

Solve the inequality
X²-|x|-12
________ is less than or equal to 0
|X|

Answer 1

Linear Algebra

We need to solve the given inequality $\frac {x^2 - |x| - 12} {|x|}\le 0$

Solution:

Given inequality

$$\frac {x^2 - |x| - 12} {|x|}\le 0$$

Multiply with $|x|$ on both sides of inequality, then we get

$$x^2 - |x| - 12 \le 0 \space and \space |x|>0$$

We can also write this as,

$$|x|^2 - |x| - 12\le0 \space and \space |x|>0$$

We can factorize this as,

$$|x|^2 - 4|x| + 3|x| - 12 \le 0$$

$$|x| ( |x| - 4) + 3 (|x| - 4) \le 0$$

$$(|x|+ 3) (|x| - 4) \le 0$$

$$(|x|- (- 3)) (|x| - 4) \le 0$$

We can write the |x| as,

$$- 3\le |x| \le 4 \space and \space |x|>0$$

But |x| should not be negative and greater than 0

So, the solution would be,

$$0< |x| \le 4$$