Find all values of x that satisfy the following equation:|x| + |x + 1| < 2.

Sure, let's solve this inequality step by step:

The absolute value function |x| splits the number line into two regions: x < 0 and x ≥ 0. Similarly, the absolute value function |x + 1| splits the number line into two regions: x < -1 and x ≥ -1. Therefore, we have three regions to consider: x < -1, -1 ≤ x < 0, and x ≥ 0.

1. For x < -1, both |x| and |x + 1| are negative, so the inequality becomes -(x) - (x + 1) < 2, which simplifies to -2x - 1 < 2. Solving for x gives x > -3/2. However, since we are considering the region x < -1, there are no solutions in this region.

2. For -1 ≤ x < 0, |x| is negative and |x + 1| is positive, so the inequality becomes -(x) + (x + 1) < 2, which simplifies to 1 < 2. This is always true, so all x in the region -1 ≤ x < 0 are solutions.

3. For x ≥ 0, both |x| and |x + 1| are positive, so the inequality becomes x + (x + 1) < 2, which simplifies to 2x + 1 < 2. Solving for x gives x < 1/2. However, since we are considering the region x ≥ 0, the solutions in this region are 0 ≤ x < 1/2.

Therefore, the solutions to the inequality |x| + |x + 1| < 2 are -1 ≤ x < 1/2.