First slide

Introduction to linear inequalities

Question

$The solution of the inequality \frac{| x + 2 | - | x |}{\sqrt{8 - x^{3}}} \geq 0 is$

Moderate

A

$[- 1, 2)$
B

$[1, 2]$
C

$[- 1, 1]$
D

$[0, \sqrt[3]{4})$

Solution

$\begin{array}{l} \frac{| x + 2 | - | x |}{\sqrt{8 - x^{3}}} \geq 0 \\ We must have 8 - x^{3} > 0 \\ \Rightarrow x^{3} < 8 \Rightarrow x < 2 \\ Also | x + 2 | - | x | \geq 0 \\ \Rightarrow | x + 2 | \geq | x | \Rightarrow (x + 2)^{2} - x^{2} \geq 0 \Rightarrow 2 (x + 1) \geq 0 \\ \Rightarrow x \geq - 1 \\ So x \in [- 1, 2) \end{array}$

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