Absolute value equations look strange at first because one equation hides two answers inside it. Once you learn the two-case method, they become routine. You can check your work with the free tools on Calc Solver.
What Absolute Value Means
The absolute value of a number is its distance from zero, so it is never negative. The absolute value of 5 is 5, and the absolute value of -5 is also 5. That is why an absolute value equation usually has two solutions: one positive case and one negative case.
The Key Idea: Two Cases
If |x| = 7, then x could be 7 or x could be -7, because both are 7 units from zero. Every absolute value equation is solved the same way: split it into two ordinary equations, one for each case.
Step 1: Isolate the Absolute Value
Before splitting, get the absolute value expression alone on one side. For |2x – 3| = 7, it is already alone, so you can split right away. If there are numbers outside the bars, undo them first, just as you would in a normal equation.
Step 2: Split Into Two Equations
Write the two cases. Case 1 keeps the inside as it is: 2x – 3 = 7. Case 2 makes the inside negative: 2x – 3 = -7. Do not change anything else when you write the second case.
Step 3: Solve Each Equation
Case 1: 2x – 3 = 7 gives 2x = 10, so x = 5. Case 2: 2x – 3 = -7 gives 2x = -4, so x = -2. The equation has two solutions: x = 5 and x = -2.
Step 4: Check Every Answer
Always put each answer back into the original equation. For x = 5: |2 times 5 – 3| = |7| = 7, correct. For x = -2: |2 times -2 – 3| = |-7| = 7, correct. Checking matters because some absolute value equations produce answers that do not work, especially when the absolute value is not isolated first.
Practice Problems to Try
Solve |x + 4| = 9, then |3x – 1| = 8, then |x – 6| = 2. For each one, write both cases, solve them, and check both answers.