<s>[INST] Polynomial inequalities Just wanted to check my maths:
a) $\displaystyle \frac{(x^3-4)}{x-2} \leq x+2$ I got that $x\leq 1$.
b) $\displaystyle \frac{(2x-3)(x+1)}{x^2} \leq -2$
I got that $\displaystyle -\frac{3}{4} \leq x \leq 1$
Right/wrong? Thanks :). I just worked on it as if it were an equals sign... so I'm not sure I'm correct.
 [/INST] In case of the 2nd one,
$\displaystyle \frac{(2x-3)(x+1)}{x^2} \leq -2$
Multiplying both sides by $x^2$ (Considering $x\ne0)$as it is +ve so no problem of sign changing,
$\Rightarrow(2x-3)(x+1)\leq -2x^2$
$\Rightarrow 2x^2-x-3\leq -2x^2$
$\Rightarrow 4x^2-x-3\leq 0$
$\Rightarrow(2x)^2-2.2x.\frac{1}{4}+1/16-1/16-3\leq0$
$\Rightarrow(2x-1/4)^2-49/16\le0$
$\Rightarrow-\frac{7}{4}\le2x-1/4\le7/4$
$\Rightarrow-3/4\le x\le 1$
This solution must be without 0.
$x\in(-3/4,1)-{0}$
</s>