If |x + 1| + |x − 2| ≥ 4, then x lies in:

Solution

Given Inequality: |x + 1| + |x − 2| ≥ 4 Identify the critical points where the expressions inside absolute values change sign. The critical points are:

• x + 1 = 0 → x = -1
• x - 2 = 0 → x = 2

These points divide the real line into three intervals: x < -1, -1 ≤ x < 2, and x ≥ 2. Analyze the inequality in each interval. Case 1: x < -1 When x < -1:

• x + 1 < 0, so |x + 1| = -(x + 1) = -x - 1
• x - 2 < 0, so |x - 2| = -(x - 2) = -x + 2

The inequality becomes: (-x - 1) + (-x + 2) ≥ 4 -2x + 1 ≥ 4 -2x ≥ 3 x ≤ -3/2 Since we're in the case x < -1, the solution in this interval is: x ≤ -3/2 Case 2: -1 ≤ x < 2 When -1 ≤ x < 2:

• x + 1 ≥ 0, so |x + 1| = x + 1
• x - 2 < 0, so |x - 2| = -(x - 2) = -x + 2

The inequality becomes: (x + 1) + (-x + 2) ≥ 4 3 ≥ 4 This is never true, so there are no solutions in this interval. Case 3: x ≥ 2 When x ≥ 2:

• x + 1 > 0, so |x + 1| = x + 1
• x - 2 ≥ 0, so |x - 2| = x - 2

The inequality becomes: (x + 1) + (x - 2) ≥ 4 2x - 1 ≥ 4 2x ≥ 5 x ≥ 5/2 Since we're in the case x ≥ 2, and 5/2 = 2.5 > 2, the solution in this interval is: x ≥ 5/2 Step 3: Combine the solutions. From Case 1: x ≤ -3/2 From Case 2: No solutions From Case 3: x ≥ 5/2 Therefore, the solution is: x ∈ (-∞, -3/2] ∪ [5/2, +∞) So the answer is x ≤ −1.5 or x ≥ 2.5.