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Question Number 166580 by mkam last updated on 22/Feb/22

	Answered by mahdipoor last updated on 22/Feb/22

	$1)$ $\Rightarrow∣ x ∣⩾∣ 2 x + 1 ∣\Rightarrow x^{2} ⩾ 4 x^{2} + 4 x + 1 \Rightarrow$ $0 ⩾ 3 x^{2} + 4 x + 1 = (3 x + 1) (x + 1) \Rightarrow$ $x \in [- 1, - 1 / 3]$ $2)$ $x - 5 = \pm (3 x - 1) \Rightarrow x = - 2, 3 / 2$ $3)$ $i) x < - 5 \Rightarrow x + 5 < 0, 3 x + 1 < 0 \Rightarrow$ $- (3 x + 1) > (x + 5) x \Rightarrow 0 > x^{2} + 8 x + 1 \Rightarrow$ $x \in (- 4 - \sqrt{15}, - 4 + \sqrt{15})$ $\Rightarrow x \in (- 4 - \sqrt{15}, - 5)$ $i i) - 1 / 3 ⩾ x > - 5 \Rightarrow x + 5 > 0, 3 x + 1 < 0 \Rightarrow$ $- (3 x + 1) < x (x + 5) \Rightarrow 0 < x^{2} + 8 x + 1 \Rightarrow$ $x \in] - 4 - \sqrt{15}, - 4 + \sqrt{15} [$ $\Rightarrow x \in \emptyset$ $i i i) x ⩾ - 1 / 3 \Rightarrow x + 5 > 0, 3 x + 1 > 0 \Rightarrow$ $(3 x + 1) < x (x + 5) \Rightarrow 0 < x^{2} + 2 x - 1 \Rightarrow$ $x \in (- 1 - \sqrt{2}, - 1 + \sqrt{2})$ $\Rightarrow x \in [- 1 / 3, - 1 + \sqrt{2})$ $\Rightarrow i i i \cup i i \cup i \Rightarrow$ $x \in (- 4 - \sqrt{15}, - 5) \cup [- 1 / 3, - 1 + \sqrt{2})$
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