Absolute Value Equations

Absolute Value Equations: how to solve them

Absolute value measures distance from 0, so an equation with (|,\cdot,|) often splits into two possible cases.

1) Isolate the absolute value

Rewrite the equation so the absolute value expression is alone on one side. For example, aim for a form like (|x-3|=5).

2) Create the two cases

If (|A|=b) and (b\ge 0), then:

• (A=b)
• (A=-b)

If the number on the other side is negative, there is no solution, because an absolute value cannot be less than 0.

3) Solve each equation

Solve both resulting equations separately. You may get one solution, two solutions, or no solution.

4) Check your answers

Substitute each solution back into the original equation. The left side should match the right side exactly.

Example check

If (|x-4|=3), then:

• (x-4=3) gives (x=7)
• (x-4=-3) gives (x=1)

Check: (|7-4|=3) and (|1-4|=3), so both are correct.

Tip

Always simplify the final answer, and write multiple solutions clearly if they exist.