Reduce to l form $ ax ^ 2 + bx + c $ in the first member, $ 0 $ in the 2nd member, then use the discussion of the sign of$T = ax^2+ bx+c $ $\Delta = b^2 - 4ac$ ------------------------------- Si $\Delta\lt 0$: $T$ a le sign of $a$ ------------------------------- Si $\Delta = 0$: $T$ has the sign of $a$, if $x$ is not equal to the root $x_0$ $ T = 0 $, if $ x $ is equal to the root $ x_0 $ ------------------------------- If $ \Delta \gt 0 $: $ T = 0 $, if $ x $ is equal to the roots $ x_1 $ or $ x_2 $ $ T $ has the sign of $ a $, if $ x $ is outside the roots $ x_1 $ and $ x_2 $ $ T $ has the sign of $ -a $, if $ x $ is between the roots $ x_1 $ and $ x_2 $ ------------------------------- -------------------------------

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$\LARGE -x^2 \lt x - 12$	$\LARGE x^2 + x -12 \gt 0$,then $\LARGE x\gt 3$ or $\LARGE x\lt -4$, $\LARGE S = ]-\infty ,-4[ \cup ]3 , +\infty [$
$\LARGE x^2 \lt 8 - 7x$	$\LARGE x^2 + 7x -8 \lt 0$ ,then $\LARGE x \gt -8$ et $ x\lt 1$, $\LARGE S = ]-8, 1[$
$\LARGE x^2 + 31x \gt -150$	$\LARGE x^2 + 31x + 150 \gt 0$ ,then $\LARGE x \gt -6$ or $\LARGE x\lt -25$, $\LARGE S = ]-\infty ,-25[ \cup ]-6 , +\infty [$
$\LARGE x^2 -3x \geq -2$	$\LARGE x^2 -3x +2 \geq 0$ ,then $\LARGE x \geq 2$ or $\LARGE x \leq 1$, $\LARGE S = ]-\infty ,1] \cup [2 , +\infty [$
$\LARGE x^2 + 31x \gt -150$	$\LARGE x^2 + 31x + 150 \gt 0$ ,then $\LARGE x \gt -6$ or $\LARGE x\lt -25 $, $\LARGE S = ]-\infty ,-25[ \cup ]-6 , +\infty [$
$\LARGE 5 - 4x \gt x^2$	$\LARGE x^2 + 4x - 5 \lt 0$ ,then $\LARGE -5 \lt x \lt 1$ , $\LARGE S = ]-5, 1[$
$\LARGE 4x(x+3) \geq -9$	$\LARGE 4x^2 + 12x +9 \geq 0$ ,then $\LARGE S = \mathbb R$
$\LARGE 5 - 4x \gt x^2$	$\LARGE x^2 + 4x - 5 \lt 0$ ,then $\LARGE S = \mathbb R \setminus \{1\}$