Answer:
-1 ≤ x ≤ 2
Explanation:
You want the solution to |x +1| +|x -2| = 3.
Graph
We find it convenient to solve these absolute value equations using a graphing calculator. When we subtract 3 from both sides, we have ...
|x +1| +|x -2| -3 = 0
The solutions will show on the graph as places where the expression has a value of 0, that is, the x-intercepts.
The left-side expression has a value of 0 for all values of x between -1 and +2, inclusive. That is, the solution is ...
-1 ≤ x ≤ 2
Algebra
The absolute value function is piecewise defined:
|x| = x . . . . for x ≥ 0
|x| = -x . . . . for x < 0
That is, the behavior of the function changes at x=0.
In the given equation the absolute value function arguments are zero at ...
x +1 = 0 ⇒ x = -1
x -2 = 0 ⇒ x = 2
These x-values divide the domain of the equation into three parts.
x < -1
In this domain, both arguments are negative, so the equation is actually ...
-(x +1) -(x -2) = 3
-2x +1 = 3
-2x = 2
x = -1 . . . . . . not in the domain
-1 ≤ x < 2
In this domain, the argument (x+1) is positive, but the argument (x-2) is negative. That means the equation is ...
(x +1) -(x -2) = 3
1 +2 = 3
True for all x in this domain.
x ≤ 2
In this domain, both arguments are positive, so the equation is ...
(x +1) +(x -2) = 3
2x -1 = 3
2x = 4
x = 2 . . . . in the domain (this point was excluded from x < 2).
The solution is -1 ≤ x ≤ 2.