1-Prove-that-a-b-c-a-b-c-2-Find-all-x-R-that-satify-the-follow-ing-inequalities-i-x-2-4-lt-5-ii-x-x-2-lt-5-Help-

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} .