What is |x-1|-|x-2|+|x+1|>|x+2|+|x|-3 ?

The solution to |x-1|-|x-2|+|x+1|>|x+2|+|x|-3 is -3<x<-1\lor-1<x<1\lor 1<x<3

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|x-1|-|x-2|+|x+1|>|x+2|+|x|-3

∣ x - 1 ∣ - ∣ x - 2 ∣ + ∣ x + 1 ∣ > ∣ x + 2 ∣ + ∣ x ∣ - 3

- 3 < x < - 1 or - 1 < x < 1 or 1 < x < 3

Interval Notation

(- 3, - 1) \cup (- 1, 1) \cup (1, 3)

Solution steps

∣ x - 1 ∣ - ∣ x - 2 ∣ + ∣ x + 1 ∣ > ∣ x + 2 ∣ + ∣ x ∣ - 3

Find positive and negative intervals

x < - 2, - 2 \leq x < - 1, - 1 \leq x < 0, 0 \leq x < 1, 1 \leq x < 2, x \geq 2

Solve the inequality for each interval

- 3 < x < - 2 or - 2 \leq x < - 1 or - 1 < x < 0 or 0 \leq x < 1 or 1 < x < 2 or 2 \leq x < 3

Merge Overlapping Intervals

- 3 < x < - 1 or - 1 < x < 1 or 1 < x < 3

\frac{x - 3}{x + 1} \leq 1

\frac{( 3 x )}{2 + 5} \leq 2

\frac{( x + 4 )}{( x + 2 )} \leq 1

∣ x + 4 ∣ > ∣ 2 x - 3 ∣

∣ 2 x - 3 ∣ > 20

What is |x-1|-|x-2|+|x+1|>|x+2|+|x|-3 ?
The solution to |x-1|-|x-2|+|x+1|>|x+2|+|x|-3 is -3<x<-1\lor-1<x<1\lor 1<x<3