First slide

Introduction to linear inequalities

Question

$The inequality {(\frac{1}{2})}^{x^{6} - 2 x^{4}} < 2^{x^{2}} is valid for x belongs to$

Moderate

A

$R$
B

$(0, \infty)$
C

$(- \infty, - 1) \cup (- 1, \infty)$
D

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Solution

$\begin{aligned} 2^{2 x^{4} - x^{6}} < 2^{x^{2}} \\ \Rightarrow 2 x^{4} - x^{6} < x^{2} \\ \Rightarrow x^{6} - 2 x^{4} + x^{2} > 0 \\ \Rightarrow x^{2} (x^{4} - 2 x^{2} + 1) > 0 \\ \Rightarrow x^{2} (x - 1)^{2} (x + 1)^{2} > 0 \end{aligned}$
$which is true always except at x = 0, 1, - 1$

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