Solve the following inequality:
${0.1}^{4 x^{2} - 2 x - 2} ⩽ {0.1}^{2 x - 3}$

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Solution

${0.1}^{4 x^{2} - 2 x - 2} \leq {0.1}^{2 x - 3}$
Taking log on both the sides,
$\Rightarrow 4 x^{2} - 2 x - 2 \geq 2 x - 3$
$\Rightarrow 4 x^{2} - 4 x + 1 \geq 0$
$\Rightarrow (x - \frac{1}{2})^{2} \geq 0$
valid for all $x$

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