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Question Number 18653 by Tinkutara last updated on 26/Jul/17

Solve the inequality, ∣x − 1∣ + ∣x + 1∣ < 4

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{inequality},\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:+\:\mathrm{1}\mid\:<\:\mathrm{4} \\ $$

Answered by ajfour last updated on 26/Jul/17

If x<−1 −(x−1)−(x+1)<4 ⇒ 2x>−4 or x> −2 hence x∈(−2 ,−1) If −1≤x<1 −(x−1)+x+1<4 ⇒ 2<4 so x∈[−1,1) If 1≤x x−1+x+1<4 ⇒ 2x<4 or x<2 x∈[1,2) Taking union of the three cases, x∈(−2,2) .

$$\mathrm{If}\:\mathrm{x}<−\mathrm{1} \\ $$ $$−\left(\mathrm{x}−\mathrm{1}\right)−\left(\mathrm{x}+\mathrm{1}\right)<\mathrm{4} \\ $$ $$\Rightarrow\:\mathrm{2x}>−\mathrm{4}\:\:\mathrm{or}\:\:\mathrm{x}>\:−\mathrm{2} \\ $$ $$\mathrm{hence}\:\mathrm{x}\in\left(−\mathrm{2}\:,−\mathrm{1}\right) \\ $$ $$\mathrm{If}\:−\mathrm{1}\leqslant\mathrm{x}<\mathrm{1} \\ $$ $$−\left(\mathrm{x}−\mathrm{1}\right)+\mathrm{x}+\mathrm{1}<\mathrm{4} \\ $$ $$\Rightarrow\:\mathrm{2}<\mathrm{4}\: \\ $$ $$\mathrm{so}\:\:\mathrm{x}\in\left[−\mathrm{1},\mathrm{1}\right) \\ $$ $$\mathrm{If}\:\mathrm{1}\leqslant\mathrm{x} \\ $$ $$\mathrm{x}−\mathrm{1}+\mathrm{x}+\mathrm{1}<\mathrm{4} \\ $$ $$\Rightarrow\:\mathrm{2x}<\mathrm{4}\:\:\mathrm{or}\:\:\:\mathrm{x}<\mathrm{2} \\ $$ $$\:\:\mathrm{x}\in\left[\mathrm{1},\mathrm{2}\right) \\ $$ $$\mathrm{Taking}\:\mathrm{union}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three}\:\mathrm{cases}, \\ $$ $$\:\:\:\:\:\mathrm{x}\in\left(−\mathrm{2},\mathrm{2}\right)\:. \\ $$

Commented byTinkutara last updated on 26/Jul/17

Thank you very much ajfour Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{ajfour}\:\mathrm{Sir}! \\ $$

Commented byajfour last updated on 26/Jul/17

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