Absolute Value Equations

The absolute value is the distance of from zero on the number line, so it is never negative. An equation like says “the quantity is a distance of from zero” — and there are two ways for that to happen: or .

That split into two cases is the whole method. The steps around it are: isolate the absolute value first, then split, then solve each linear (or quadratic) piece with the methods you already know. Three special cases — equal to zero, equal to a negative, and absolute value equal to absolute value — are worth memorizing.

It helps to keep the definition in view. Formally,

When you write you are assuming the inside came out non‑negative; when you write you are assuming it came out negative and the bars flipped its sign. Those are the only two possibilities, which is exactly why an isolated with produces two equations and never three. Every technique in this lesson is just bookkeeping around that one fact.

The Absolute Value Equation Formula

is whatever is inside the absolute value bars. When there are two cases; gives the single equation ; gives no solution.

The Method

1. Isolate the absolute value expression — get alone on one side.
2. Check the right side.
   • Positive: split into and .
   • Zero: solve the single equation .
   • Negative: no solution, .
3. Solve each resulting equation.
4. Check if the other side contains the variable.

Special Case: Equal to Zero

One solution — the two cases collapse because and are the same.

Special Case: Equal to a Negative

Distance is never negative.

Special Case: Absolute Value Equals Absolute Value

The Squaring Alternative for |A| = |B|

Because for every real number, the equation is equivalent to . Squaring both sides removes the bars entirely:

The zero‑product step recovers the same two cases, and , so squaring is not a new method — it is a check on the split. Squaring is only safe when both sides are absolute values (or both are known to be non‑negative). If one side contains a bare variable expression such as , squaring can introduce extraneous roots, so the isolate‑and‑split method with a final check is the reliable route there.

When the Split Produces a Quadratic

Nothing requires the inside of the bars to be linear. still splits into two equations, they just happen to be quadratic:

The first gives . The second gives , whose discriminant is , so it contributes no real solutions. Treat each case with whatever quadratic tool fits — factoring, the quadratic formula, or completing the square — and discard any case with no real roots.

Worked Example A: The Basic Split

Solve .

Step 1 — already isolated. Right side is positive, so split:

Step 2 — solve each:

Answer: or . Both are a distance from .

Worked Example B: Isolate First

Solve .

Step 1 — isolate the absolute value:

Step 2 — positive right side, split:

Step 3 — solve each:

Answer: or .

Worked Example C: Variable on Both Sides

Solve .

Step 1 — split:

Step 2 — solve each:

Step 3 — check both (the right side must be ):

• : and . ✓
• : but . . ✗ extraneous

Answer: only.

Worked Example D: Absolute Value Equals Absolute Value

Solve .

Split into and :

Solve each:

Check: : . ✓ : . ✓

Answer: or .

Worked Example E: A Quadratic Inside the Bars

Solve .

Step 1 — already isolated, right side positive, split:

Step 2 — solve each case as a quadratic:

Step 3 — check the real candidates in the original equation:

• : . ✓
• : . ✓

Answer: or .

Common Mistakes to Avoid

• Splitting before isolating. is not or ; first get .
• Only writing one case. with always produces two equations.
• Reporting an answer when the right side is negative. has no solution — do not split it.
• Forgetting the check when the variable is on both sides. A candidate can make the right side negative, which invalidates it.
• Distributing the negative to only one term. In case two, is , not .
• Dropping the ”” single solution. has exactly one solution, not zero and not two.

Where This Shows Up Later

• Absolute Value Inequalities. and use the same “inside is a distance” idea, turned into compound inequalities.
• Distance on the number line and in the plane. is the distance between and ; equations set that distance to a value.
• Piecewise functions. is the piecewise function for and for ; the two cases here are those two pieces.
• Error bounds and tolerances. “The measurement is within of ” is at the boundary.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve two ways — by splitting and by squaring — and confirm they match.

Quick Reference

Each split case is a Linear Equation (or occasionally a quadratic). The inequality companion is Absolute Value Inequalities. More Algebra lessons are available too.