Be calm then solve ∣((x^2 +7x−8)/(x+3))∣≥ 2


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Question Number 178612 by Acem last updated on 19/Oct/22

$B e c a l m t h e n s o l v e ∣ \frac{x^{2} + 7 x - 8}{x + 3} ∣⩾ 2$
	Answered by cortano1 last updated on 19/Oct/22

	$x \neq - 3$ $∣ x^{2} + 7 x - 8 ∣⩾∣ 2 x + 6 ∣$ $(x^{2} + 7 x - 8 + 2 x + 6) (x^{2} + 7 x - 8 - 2 x - 6) ⩾ 0$ $(x^{2} + 9 x - 2) (x^{2} + 5 x - 14) ⩾ 0$ $[{(x + \frac{9}{2})}^{2} - \frac{89}{4} -] (x + 7) (x - 2) ⩾ 0$ $(x + \frac{9 + \sqrt{89}}{2}) (x + (\frac{9 - \sqrt{89}}{2})) (x + 7) (x - 2) ⩾ 0$
	Commented by Acem last updated on 19/Oct/22

	$G o o d s i r! t r y t o f i n d t h e r e q u i r e d x v a l u e s$ $T i p : ∣ P ∣⩾ d; d ⩾ 0 \Rightarrow$ $E i t h e r P ⩾ d O r P ⩽ - d$
	Commented by cortano1 last updated on 19/Oct/22

	$i {don}^{'} t try it$
	Commented by Acem last updated on 19/Oct/22

	$O k f r i e n d, y o u c a n c h e c k i t b e l o w$
	Answered by Acem last updated on 19/Oct/22

	$E i t h e r \frac{x^{2} + 7 x - 8}{x + 3} ⩾ 2 ∣_{A} O r \frac{x^{2} + 7 x - 8}{x + 3} ⩽ - 2 ∣_{B}$ $A : \frac{(x - 2) (x + 1)}{x + 3} ⩾ 0$ $x - \infty - 7 - 3 2 + \infty$ $(x - 2) (x + 1) + 0 - - 0 +$ $x + 3 - - 0 + +$ $F r a c . - 0 + ∥ - 0 +$ $E i t h e r x \in [- 7, - 3 [\cup [2, + \infty [$ $B : \frac{{(x + \frac{9}{2})}^{2} - \frac{89}{4}}{x + 3} ⩽ 0$ $x = \frac{- 9 - \sqrt{89}}{2} ≅ - 9.21 o r x = \frac{- 9 + \sqrt{89}}{2} ≅ 0.21$ $x - \infty x_{1} - 3 x_{2} + \infty$ $(x - 2) (x + 1) + 0 - - 0 +$ $x + 3 - - 0 + +$ $F r a c . - 0 + ∥ - 0 +$ $O r x \in] - \infty, \frac{- 9 - \sqrt{89}}{2}] \cup] - 3, \frac{- 9 + \sqrt{89}}{2}]$
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