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Formation of Algebraic Expressions

Solve the fol...

Question

Solve the following inequality:
$x^{2} - | 5 x + 8 | > 0.$

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Solution

$x^{2} - | 5 x + 8 | > 0$

$| 5 x + 8 | = 5 x + 8$

$5 x + 8 \geq 0 : x \geq - \frac{8}{5}$

$| 5 x + 8 | = - (5 x + 8)$

$x < - \frac{8}{5}$

$x^{2} - (5 x + 8) > 0$

$x^{2} - 5 x - 8 > 0$

$(x - \frac{5 + \sqrt{57}}{2}) (x - \frac{5 - \sqrt{57}}{2}) > 0$

$(x < \frac{5 - \sqrt{57}}{2} o r x > \frac{5 + \sqrt{57}}{2}) a n d (x \geq - \frac{8}{5})$

$- \frac{8}{5} \leq x < \frac{5 - \sqrt{57}}{2} o r x > \frac{5 + \sqrt{57}}{2}$

$x < - \frac{8}{5} o r (- \frac{8}{5} \leq x < \frac{5 - \sqrt{57}}{2} o r x > \frac{5 + \sqrt{57}}{2})$

$x < \frac{5 - \sqrt{57}}{2} o r x > \frac{5 + \sqrt{57}}{2}$

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