Absolute Value Inequality

The above inequality can be rewritten as:

$(\frac{x^2+1}{x})(|x|-17)(|x|+5) \le 0$

If we first examine the inequality over the interval $(-17,0)$ , the above factors yield:

$\frac{x^2+1}{x} =$ Negative, $|x|-17 =$ Negative, $|x|+5 =$ Positive

which violates the inequality as the product is positive overall. If we now examine the interval $(0, 17]$ , the factors yield:

$\frac{x^2+1}{x} =$ Positive, $|x|-17 =$ Non-positive, $|x|+5 =$ Positive

which satisfies the inequality as the overall product is either negative or zero. Hence, there are $\boxed{17}$ such integral values of $x$ .

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