The number of integral values of x that satisfying the inequality 2-x23x-35x-1x2-3x-4-0 is

(2 x^2)^3(x 3)^5(x+1)(x^2 3x 4)≥0 ⇒(x^2 2)^3(x 3)^5(x+1)(x+1)(x 4)≤0 ⇒(x+√(2))^3(x √(2))^3(x 3)^5(x+1)^2(x 4)≤0 ⇒ x∈[ √(2), 1)∪( 1,√(2)]∪[3,4) Integers are 0,1, ...

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The number of...

Question

The number of integral value(s) of $x$ that satisfying the inequality $\frac{(2 - x^{2})^{3} (x - 3)^{5}}{(x + 1) (x^{2} - 3 x - 4)} \geq 0$ is

3.00

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Solution

$\frac{(2 - x^{2})^{3} (x - 3)^{5}}{(x + 1) (x^{2} - 3 x - 4)} \geq 0$
$\Rightarrow \frac{(x^{2} - 2)^{3} (x - 3)^{5}}{(x + 1) (x + 1) (x - 4)} \leq 0$
$\Rightarrow \frac{(x + \sqrt{2})^{3} (x - \sqrt{2})^{3} (x - 3)^{5}}{(x + 1)^{2} (x - 4)} \leq 0$
$\Rightarrow x \in [- \sqrt{2}, - 1) \cup (- 1, \sqrt{2}] \cup [3, 4)$
Integers are $0, 1, 3$

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