Resolver ((∣x − 1∣ + ∣x + 2∣)/(∣x∣ − 1)) ≤ 3


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Question Number 217389 by Simurdiera last updated on 12/Mar/25

$Resolver$ $\frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} \leq 3$
	Answered by A5T last updated on 13/Mar/25

	$when - 2 ⩽ x < - 1; ∣ x - 1 ∣= 1 - x, ∣ x + 2 ∣= x + 2 \land ∣ x ∣= - x$ $\Rightarrow \frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} ⩽ 3 \Rightarrow \frac{1 - x + x + 2}{- x - 1} ⩽ 3$ $\Rightarrow 3 ⩽ - 3 x - 3 \Rightarrow x ⩽ - 2 but - 2 ⩽ x < - 1 \Rightarrow x = - 2$ $when - 1 < x < 0$ $\Rightarrow∣ x ∣= - x, ∣ x - 1 ∣= 1 - x and ∣ x + 2 ∣= x + 2$ $\Rightarrow \frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} ⩽ 3 \Rightarrow \frac{1 - x + x + 2}{- x - 1} ⩽ 3$ $- x - 1 < 0 \Rightarrow 3 ⩾ 3 (- x - 1) \Rightarrow 2 ⩾ - x$ $\Rightarrow x ⩾ - 2 but - 1 < x < 0 \Rightarrow x \in (- 1, 0)$ $x = 0 makes it true$ $when 0 < x < 1; ∣ x - 1 ∣= 1 - x, ∣ x + 2 ∣= x + 2 \land ∣ x ∣= x$ $\Rightarrow \frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} ⩽ 3 \Rightarrow \frac{1 - x + x + 2}{x - 1} ⩽ 3$ $x - 1 < 0 \Rightarrow 3 ⩾ 3 (x - 1)$ $\Rightarrow x ⩽ 1 but 0 < x < 1 \Rightarrow x \in (0, 1)$ $when x > 1; ∣ x - 1 ∣= x - 1, ∣ x + 2 ∣= x + 2 \land ∣ x ∣= x$ $\Rightarrow \frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} ⩽ 3 \Rightarrow \frac{x - 1 + x + 2}{x - 1} ⩽ 3$ $\Rightarrow 2 x + 1 ⩽ 3 x - 3$ $\Rightarrow x ⩾ 4 but x > 1 \Rightarrow x \in [4, \infty)$ $when x < - 2; ∣ x - 1 ∣= 1 - x, ∣ x + 2 ∣= - x - 2 \land ∣ x ∣= - x$ $\Rightarrow \frac{∣ x - 1 ∣ + ∣ x + 2 ∣}{∣ x ∣ - 1} ⩽ 3 \Rightarrow \frac{1 - x - x - 2}{- x - 1} ⩽ 3$ $\Rightarrow - 2 x - 1 ⩽ - 3 x - 3$ $\Rightarrow x ⩽ - 2 but x < - 2 \Rightarrow x \in (- \infty, - 2)$ $\Rightarrow x \in (- \infty, - 2] \cup (- 1, 1) \cup [4, \infty)$
	Commented by Simurdiera last updated on 12/Mar/25

	$G r a c i a s \underset{―}{\underset{⏟}{⋚}}$
	Commented by Hanuda354 last updated on 13/Mar/25

	$x \in (- \infty, - 2] \cup (- 1, 1) \cup [4, \infty)$
	Answered by mehdee7396 last updated on 12/Mar/25

	$x = 0, 1, - 2$ $i f x ⩽ - 2 (1) \Rightarrow \frac{- x + 1 - x - 2}{- x - 1} - 3 ⩽ 0$ $\frac{x + 2}{- x - 1} ⩽ 0 \Rightarrow x ⩽ - 2 o r x > - 1 (2)$ $(1) \cap (2) \Rightarrow x ⩽ - 2 (i)$ $i f - 2 ⩽ x ⩽ 0 (1) \Rightarrow \frac{- x + 1 + x + 2}{- x - 1} - 3 ⩽ 0$ $\frac{3 x + 6}{- x - 1} ⩽ 0 \Rightarrow x ⩽ - 2 o r x > - 1 (2)$ $(1) \cap (2) \Rightarrow x \in {- 2} \cup (- 1, 0] (i i)$ $i f 0 ⩽ x < 1 (1) \Rightarrow \frac{- x + 1 + x + 2}{x - 1} - 3 ⩽ 0$ $\frac{- 3 x + 6}{x - 1} ⩽ 0 \Rightarrow x < 1 o r x ⩾ 2 (2)$ $(1) \cap (2) \Rightarrow 0 ⩽ x < 1 (i i i)$ $i f 1 < x (1) \Rightarrow \frac{x - 1 + x + 2}{x - 1} - 3 ⩽ 0$ $\frac{- x + 4}{x - 1} ⩽ 0 \Rightarrow x < 1 o r x ⩾ 4 (2)$ $(1) \cap (2) \Rightarrow x = 1 o r x ⩾ 4 (i v)$ $(i) \cup (i i) \cup (i i i) \cup (i v) = (- \infty, - 2] \cup (- 1, 1) \cup [4, \infty)$
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