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Integration of Irrational Algebraic Fractions - 1

Solve the fol...

Question

Solve the following inequality:
$1 < \frac{3 x^{2} - 7 x + 8}{x^{2} + 1} ⩽ 2$

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Solution

$\frac{3 x^{2} - 7 x + 8}{x^{2} + 1} > 1$
$3 x^{2} - 7 x + 8 > x^{2} + 1$
$2 x^{2} - 7 x + 7 > 0$
As $△ < 0$ $2 x^{2} - 7 x + 7 > 0$
$\frac{3 x^{2} - 7 x + 8}{x^{2} + 1} \leq 2$
$3 x^{2} - 7 x + 8 \leq 2 x^{2} + 2$
$x^{2} - 7 x + 6 \leq 0$
$(x - 6) (x - 1) \leq 0$
$x \in (1, 6)$

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