inequality

Solve the inequality:

(|x|-2)² >= x+4

(|x|-2)² >= x+4

Or, (|x|²-4|x|+4) >= x+4

Or, (x²-4|x|+4) >= x+4 ………………because |x|²= x²

Substituting |x| = +x and |x|= -x and solving the above inequality

For, |x| = +x

x²-4x+4 >= x+4

Or, x²-5x>= 0

Or, x(x – 5)>=0,

so, x >= 0 or, x >= 5,  OR x <= 0 or, x <= 5

so, general solution x >= 5, or x <= 0        --------------------(1)

For, |x| = -x

x²+4x+4 >= x+4

Or, x²+3x>= 0

Or, x(x +3)>=0,

so, x >= 0 or, x >= -3, OR x <= 0 or, x <= -3

so, general solution x >= 0, or x <= -3  -------------------(2)

on combining (1) and (2), we have

x <= -3  or x >= 5