Absolute Value Equation Calculator: Solve |ax+b|=c with Steps

Q: What happens when c equals zero?

The two cases merge into one. |ax + b| = 0 means ax + b = 0, so there is a single solution. For example, |2x - 6| = 0 gives 2x = 6 and x = 3.

Q: What if a number sits outside the bars, like 2|x - 1| = 10?

Isolate the bars first by dividing both sides by 2, giving |x - 1| = 5. Then split as usual: x - 1 = 5 gives x = 6, and x - 1 = -5 gives x = -4.

Q: Why can an answer be a fraction?

Because the final division does not always come out whole. A result like -17/3 is exact and correct. You can also write it as a decimal, about -5.6667, but the fraction keeps full accuracy.

How to use the absolute value equation calculator

Enter an absolute value equation such as |2x-3|=7. The calculator splits it into the two cases ax+b=c and ax+b=-c, solves each, and checks both answers in the original equation.

Worked example

|2x-3|=7 splits into 2x-3=7 (so x=5) and 2x-3=-7 (so x=-2). Checking: |2(5)-3|=|7|=7 and |2(-2)-3|=|-7|=7, so both solutions are valid.

For another fully worked example with both cases checked, see solving absolute value equations step by step.

Frequently asked questions

What does |x| mean?

|x| is the distance of x from zero on the number line, so |x| is always zero or positive. That is why |ax+b|=c has two solutions when c is positive.

What if c is negative?

An absolute value can never be negative, so |ax+b|=c with c<0 has no solution.

How the two-case method works

An absolute value removes a sign, so the expression inside the bars could have started out positive or negative. Solving means covering both possibilities. First isolate the bars on one side of the equation, then split the problem into two plain linear equations and solve each one on its own.

|ax + b| = c means ax + b = c or ax + b = -c, when c is zero or positive

The two answers are usually different, and both need a quick check in the original equation to confirm no arithmetic slip has crept in.

Example: Solve |3x + 4| = 13 for a delivery distance check

Step 1, Split the cases: Write the two equations hidden inside the bars: 3x + 4 = 13 and 3x + 4 = -13.

Step 2, Solve the first case: Subtract 4 to get 3x = 9, then divide by 3, so x = 3.

Step 3, Solve the second case: Subtract 4 to get 3x = -17, then divide by 3, so x = -17/3, about -5.6667.

Step 4, Check both answers: |3(3) + 4| = |13| = 13 and |3(-17/3) + 4| = |-13| = 13, so both solutions are correct.

Common mistakes

Solving only one case: Stopping after ax + b = c misses the second solution that comes from the negative case.
Splitting before isolating: If the equation is |3x + 4| + 2 = 15, subtract 2 first so the bars stand alone before you split.
Dropping the negative sign: The second case uses -c on the right side, not c with a changed sign inside the bars.

When to use this calculator

Use it whenever a distance, a tolerance, or a difference from a target value is written with bars, such as a part that must measure within a set amount of a target size. It is also a fast way to check homework, since the tool is free to use and shows both cases.