Exercise 320Algebra › Equations and radicalsHigh schoolAdded 31 Oct 2026

An equation with absolute values

#absolute value#equations

Statement

Solve in $\mathbb{R}$ the equation $|x-1|+|x+2|=5$ .

Hint

Split $\mathbb{R}$ into three intervals according to the signs of $x-1$ and $x+2$ .

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Case $x\ge1$ . The equation becomes $(x-1)+(x+2)=5$ , that is $2x+1=5$ , so $x=2$ , which is in the interval.

Case $-2\le x<1$ . The equation becomes $(1-x)+(x+2)=3=5$ , which is impossible.

Case $x<-2$ . The equation becomes $(1-x)+(-x-2)=5$ , that is $-2x-1=5$ , so $x=-3$ , which is in the interval.

The solution set is $\{-3,\,2\}$ .