What does absolute value mean?
The absolute value |u| is the distance from u to zero on the real number line, so it is never negative. Both 7 and −7 have absolute value 7. That symmetry explains why |A| = B usually splits into A = B or A = −B when B is positive.
If the target B is zero, only A = 0 is needed because the two branches are identical. If B is negative, there is no real solution: a distance cannot equal a negative number. This classification should happen before solving the inner linear expression.
How to solve an absolute value equation step by step
The absolute-value bars must be isolated before the case split. If a coefficient or constant remains outside, undo those operations first while preserving equality.
- Simplify both sides and isolate the entire absolute-value expression.
- Inspect the target: negative means no real solution; zero creates one linear case; positive creates two cases.
- For a positive target, solve the positive branch A = B.
- Solve the negative branch A = −B as a separate linear equation.
- Substitute every candidate into the original absolute-value equation and remove any invalid value.
Keep the negative branch grouped. Writing A = −B means the complete right-hand expression changes sign, not just one term inside A. The calculator labels each branch so the two paths remain easy to audit.
Worked example: |2x − 3| = 7
Because 7 is positive, create two cases. Positive branch: 2x − 3 = 7, so 2x = 10 and x = 5. Negative branch: 2x − 3 = −7, so 2x = −4 and x = −2.
Check x = 5: |2(5) − 3| = |7| = 7. Check x = −2: |2(−2) − 3| = |−7| = 7. Both candidates satisfy the original equation, so the complete solution set is {−2, 5}.
Why the absolute-value term must be isolated first
For 2|x − 1| + 3 = 11, splitting immediately would use the wrong target. Subtract 3 to obtain 2|x − 1| = 8, then divide by 2 to get |x − 1| = 4. Only then create x − 1 = 4 or x − 1 = −4, producing x = 5 or x = −3.
Operations outside the bars affect the distance value, while terms inside define what is being measured. Isolating the bars separates those roles and makes the case logic valid.
Two solutions, one solution, or no solution
A positive target can produce two distinct solutions when the inner expression is genuinely linear. A zero target produces one solution because the positive and negative cases coincide. A negative target produces no real solution. Degenerate inner expressions can also change the count, which is why classification matters.
Absolute-value inequalities use a related distance idea but require interval reasoning and are outside this focused equation workflow. Do not replace = with < or > and assume the same two-equation split.
Common absolute-value equation mistakes
Stopping after the positive branch loses a valid solution. Splitting before isolating the bars uses the wrong distance. Accepting a negative target ignores the definition of absolute value. Finally, a candidate must be checked in the original equation because later transformations may hide a transcription mistake.
Use the branch headings as a checklist: target classified, positive case solved, negative case solved, and both candidates verified.