When solving an inequality involving an absolute value, what is the standard approach?

• A. Isolate inside the absolute value and solve the two cases
• B. Square both sides
• C. Ignore the absolute value and solve as a normal inequality
• D. Replace with its square root

Answer: (A)

The standard way to solve inequalities with absolute value is to isolate what’s inside the absolute value, then consider the two possibilities that the absolute value represents. Since |u| is the distance of u from zero, the inequality is really about where u lies relative to zero.

If you have |u| ≤ a with a ≥ 0, you must have -a ≤ u ≤ a. If you have |u| ≥ a, you must have u ≤ -a or u ≥ a. This split comes from the fact that the absolute value is nonnegative and captures both positive and negative versions of the inner expression.

For example, solve |2x - 3| ≤ 5. Let u = 2x - 3. Then -5 ≤ u ≤ 5, so -5 ≤ 2x - 3 ≤ 5. Adding 3 gives -2 ≤ 2x ≤ 8, and dividing by 2 yields -1 ≤ x ≤ 4.

If you instead try squaring both sides or ignoring the absolute value, you can easily miss solutions or include invalid ones, because those methods don’t faithfully reflect the two-direction nature of the inside expression that the absolute value encodes.