Solve the inequality |2x - 3| > 1.

The solution to the inequality |2x - 3| > 1 is x < 1 or x > 2.

To solve this inequality, we need to consider two cases: when 2x - 3 is positive and when it is negative.

Case 1: 2x - 3 > 0
In this case, the inequality becomes 2x - 3 > 1, which simplifies to 2x > 4 and x > 2.

Case 2: 2x - 3 < 0
In this case, the inequality becomes -(2x - 3) > 1, which simplifies to -2x + 3 > 1. Solving for x, we get x < 1.

Therefore, the solution to the inequality is x < 1 or x > 2. We can check this by plugging in values within and outside of this range into the original inequality. For example, when x = 0, |2x - 3| = 3, which is greater than 1. When x = 1.5, |2x - 3| = 0.5, which is not greater than 1. When x = 3, |2x - 3| = 3, which is greater than 1.