1) Prove that: ∣a+b+c∣≥∣a∣−∣b∣−∣c∣ 2) Find all x∈R that satify the follow− ing inequalities i) ∣x^2 −4∣<5 ii) ∣x∣+∣x+2∣<5 Help!


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Question Number 193137 by Mastermind last updated on 04/Jun/23

$1) Prove that :$ $∣ a + b + c ∣⩾∣ a ∣ - ∣ b ∣ - ∣ c ∣$ $2) Find all x \in R that satify the follow -$ $ing inequalities$ $i) ∣ x^{2} - 4 ∣< 5$ $ii) ∣ x ∣ + ∣ x + 2 ∣< 5$ $Help!$
	Answered by aba last updated on 04/Jun/23

	$∣ a ∣=∣ a + b + c - (b + c) ∣⩽∣ a + b + c ∣ + ∣ b + c ∣ (1)$ $∣ b + c ∣⩽∣ b ∣ + ∣ c ∣ \Rightarrow - ∣ b ∣ - ∣ c ∣⩽ - ∣ b + c ∣ (2)$ $(1) + (2) \Rightarrow∣ a ∣ - ∣ b ∣ - ∣ c ∣⩽∣ a + b + c ∣ ✓$
	Answered by Rajpurohith last updated on 06/Jun/23

	$1) 0 ⩽∣ a ∣=∣ a + b + c - b - c ∣⩽∣ a + b + c ∣ + ∣ b ∣ + ∣ c ∣$ $\Rightarrow∣ a ∣ - ∣ b ∣ - ∣ c ∣⩽∣ a + b + c ∣$ $2)$ $(i) T w o c a s e s a r i s e$ $∙ x^{2} ⩽ 4$ $\Rightarrow∣ x^{2} - 4 ∣= 4 - x^{2} < 5 \Rightarrow - 1 ⩽ x^{2} w h i c h i s a l w a y s t r u e .$ $s o x^{2} ⩽ 4 i s f e a s i b l e i . e, ∣ x ∣⩽ 2 .$ $∙ x^{2} > 4 \Rightarrow∣ x^{2} - 4 ∣= x^{2} - 4 < 5$ $\Rightarrow x^{2} < 9 \Rightarrow∣ x ∣< 3$ $h e n c e t h e f i n a l r e g i o n o r s o l u t i o n i s ∣ x ∣< 3 .$ $(ii) ∙ S u p p o s e x ⩾ 0$ $t h e n ∣ x ∣= x a n d ∣ x + 2 ∣= x + 2$ $s o t h e i n e q u a l i t y b e c o m e s$ $x + x + 2 < 5$ $\Rightarrow 2 x + 2 < 5 \Rightarrow x < \frac{3}{2} . H e n c e 0 ⩽ x < \frac{3}{2} i s a r e g i o n .$ $∙ s u p p o s e - 2 ⩽ x < 0 \Rightarrow x + 2 ⩾ 0$ $\Rightarrow∣ x ∣= - x a n d ∣ x + 2 ∣= x + 2$ $s o t h e i n e q u a l i t y b e c o m e s - x + x + 2 < 5 w h i c h i s t r u e .$ $s o - 2 ⩽ x < 0 i s a l s o a f e a s i b l e r e g i o n .$ $∙ s u p p o s e x < - 2 \Rightarrow∣ x + 2 ∣= - x - 2 a n d ∣ x ∣= - x$ $s o t h e i n e q u a l i t y b e c o m e s$ $- x - x - 2 < 5 \Rightarrow - 2 x < 7$ $\Rightarrow x > - \frac{7}{2}$ $h e n c e t h e f e a s i b l e r e g i o n o f t h e i n e q u a l i t y i s$ $- \frac{7}{2} < x < \frac{3}{2} .$
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