Find the number of all the integral solutions of the inequality $\frac{(x^{2} + 2) (\sqrt{x^{2} - 16})}{(x^{4} + 2) (x^{2} - 9)} \leq 0$

1

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Solution

The correct option is D $4$
$\frac{(x^{2} + 2) \sqrt{x^{2} - 16}}{(x^{4} + 2) (x^{2} - 9)} \leq 0 x^{2} + 2 > 0, \sqrt{x^{2} - 16} > 0, x^{4} + 2 > 0 \frac{1}{x^{2} - 9} \leq 0 \frac{1}{(x - 3) (x + 3)} \leq 0 x \in {- 2, - 1, 0, 1, 2} x^{2} - 16 > 0 x^{2} > 16$
$x^{2} - 9 < 0$ and $x^{2} - 16 > 0$ cant be possible at a time.
$∴$ Given equation is valid only at $x^{2} - 16 = 0$
$∴ x = \pm 4$

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