Find-all-x-R-that-satisfy-the-following-inequalities-a-4x-3-11-b-x-2-gt-x-1-c-x-x-2-2-x-8-

Question Number 193079 by Mastermind last updated on 03/Jun/23

Find all x \in R that satisfy the following

inequalities :

a) ∣ 4 x - 3 ∣⩽ 11

b) ∣ x - 2 ∣>∣ x + 1 ∣

c) ∣ x ∣ + ∣ x + 2 ∣ + ∣ 2 - x ∣⩽ 8

Commented by MM42 last updated on 04/Jun/23

a) - 11 ⩽ 4 x - 3 ⩽ 11 \Rightarrow - 8 ⩽ 4 x ⩽ 14 \Rightarrow - 2 ⩽ x ⩽ 3.5

b) i f x ⩽ - 1 \Rightarrow - x + 2 > - x - 1 \Rightarrow 2 > - 1 i s a l w a y s c o r r e c t \Rightarrow x ⩽ - 1 (i)

i f - 1 ⩽ x ⩽ 2 \Rightarrow - x + 2 > x + 1 \Rightarrow x < 0.5 \Rightarrow - 1 ⩽ x < 0.5 (i i)

i f 2 ⩽ x \Rightarrow x - 2 > x + 1 \Rightarrow - 2 > 1 \Rightarrow i s i n c o r r e c t

\Rightarrow a n s = (i) \cup (i i) = (- \infty, 0.5)

c) x ⩽ - 2 \Rightarrow - x - x - 2 + 2 - x ⩽ 8 \Rightarrow x ⩾ - \frac{8}{3} \Rightarrow - \frac{8}{3} ⩽ x ⩽ - 2 (i)

- 2 ⩽ x ⩽ 0 \Rightarrow - x + x + 2 + 2 - x ⩽ 8 \Rightarrow x ⩾ - 4 \Rightarrow - 2 ⩽ x ⩽ 0 (i i)

0 ⩽ x ⩽ 2 \Rightarrow x + x + 2 + 2 - x ⩽ 8 \Rightarrow x ⩽ 4 \Rightarrow 0 ⩽ x ⩽ 2 (i i i)

2 ⩽ x \Rightarrow x + x + 2 - 2 + x ⩽ 8 \Rightarrow x ⩽ \frac{8}{3} \Rightarrow 2 ⩽ x ⩽ \frac{8}{3} (i v)

a n s = (i) \cup (i i) \cup (i i i) \cup (i v) = [- \frac{8}{3}, \frac{8}{3}]

Commented by Subhi last updated on 03/Jun/23

(C) ∣ a + b + c ∣⩽∣ a ∣ + ∣ b ∣ + ∣ c ∣

∣ x ∣ + ∣ x + 2 ∣ + ∣ x - 2 ∣⩾∣ x + x + x + 2 - 2 ∣=∣ 3 x ∣

↓

∣ 2 - x ∣

∣ 3 x ∣⩽ 8

\frac{- 8}{3} ⩽ x ⩽ \frac{8}{3}

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