Solve Absolute Value Equations- Methods and Examples
What Absolute Value Equations Actually Are
An absolute value equation contains a variable wrapped in absolute value bars, like |x - 3| = 7. The absolute value of a number is its distance from zero on the number line. Distance is always positive, so |−5| = 5 and |5| = 5.
The equation |x| = a means x is a units from zero. That gives you two possible solutions: x = a or x = −a. This is the foundation for solving every absolute value equation you'll encounter.
The Core Rule You Need to Memorize
For any equation |A| = B where B ≥ 0:
- If B > 0, then A = B or A = −B
- If B = 0, then A = 0 only
- If B < 0, there is no solution
That's it. Everything else is applying this rule correctly.
Method 1: The Two-Case Approach
Break the equation into two separate equations based on the definition of absolute value.
Example: Solve |x - 3| = 7
Case 1: x − 3 = 7 → x = 10
Case 2: x − 3 = −7 → x = −4
Check both: |10 − 3| = 7 ✓ and |−4 − 3| = 7 ✓
Both solutions check out. Your answer is x = 10 or x = −4.
Example: Solve |2x + 1| = 5
Case 1: 2x + 1 = 5 → 2x = 4 → x = 2
Case 2: 2x + 1 = −5 → 2x = −6 → x = −3
Verify: |2(2) + 1| = 5 ✓ and |2(−3) + 1| = 5 ✓
Solutions: x = 2 or x = −3
Method 2: The Graphical Method
Rewrite the equation so the absolute value is isolated on one side. Then graph the left side (a V-shaped graph) and the right side (a horizontal line). The intersection points are your solutions.
For |x − 2| = 4, graph y = |x − 2| and y = 4. The V-shape crosses y = 4 at x = −2 and x = 6.
This method works great when you're asked to solve "using a graph" or when you want a visual check of your algebraic answer.
Method 3: Absolute Value Equals Zero
When |A| = 0, there's only one solution: A = 0. No need for two cases.
Example: Solve |3x + 6| = 0
3x + 6 = 0 → 3x = −6 → x = −2
Only one solution. |3(−2) + 6| = |−6 + 6| = 0 ✓
No Solution Situations
If the right side is negative, there's no solution. Absolute value cannot equal a negative number.
Example: |x + 4| = −3
No solution. The left side is always ≥ 0, but the right side is −3. Impossible.
Example: |2x − 1| = −5
No solution. Same reason.
Equations with Variables on Both Sides
Sometimes you'll get |A| = B where B also contains the variable. You still use two cases, but you'll need to solve each inequality or equation carefully.
Example: |x + 2| = 3x − 4
Case 1: x + 2 = 3x − 4 → 2 + 4 = 3x − x → 6 = 2x → x = 3
Case 2: x + 2 = −(3x − 4) → x + 2 = −3x + 4 → x + 3x = 4 − 2 → 4x = 2 → x = 0.5
Critical check: The right side 3x − 4 must be ≥ 0 for the original equation to make sense.
For x = 3: 3(3) − 4 = 5 ≥ 0 ✓
For x = 0.5: 3(0.5) − 4 = −2.5 < 0 ✗
Only x = 3 is valid. x = 0.5 gets discarded.
Comparing the Three Methods
| Method | Best For | Pros | Cons |
|---|---|---|---|
| Two-Case Algebra | Most standard problems | Always works, straightforward | Can forget to check solutions |
| Graphical | Visual learners, checking work | Shows all solutions at once | Less precise without technology |
| Zero Case | Equations equal to zero | Fast, one solution only | Limited to specific cases |
How to Solve Any Absolute Value Equation
Step 1: Isolate the absolute value expression on one side. Get |A| alone.
Step 2: Identify what B equals on the other side.
Step 3: If B < 0 → No solution. If B = 0 → One solution (A = 0). If B > 0 → Two cases.
Step 4: For two cases, write A = B and A = −B. Solve each.
Step 5: Check every solution in the original equation. Discard any that don't work.
Common Mistakes That Will Cost You Points
- Forgetting to set up the negative case — always write both A = B and A = −B
- Not checking solutions — always plug back in
- Solving |A| = B where B is also an expression and forgetting to verify B ≥ 0
- Dropping absolute value bars too early
- Solving for the opposite value instead of the variable
Practice Problems
1. |x − 5| = 3
Solutions: x = 8 or x = 2
2. |4x + 2| = 10
Solutions: x = 2 or x = −3
3. |x/2 − 3| = 5
Solutions: x = 16 or x = −4
4. |x² − 9| = 0
Solution: x = 3 or x = −3
Absolute value equations aren't complicated once you internalize the two-case rule. Practice setting them up, solve both cases, and always verify your answers. That's the entire game.