Absolute Value Equation Calculator

What this calculator does

So, you have an equation with an absolute value in it and you want to solve for x. This tool does that, and it shows its working step by step. You give it the coefficients of a general absolute value equation, and it isolates the absolute value, splits the problem into two cases, solves each, and reports the valid solutions.

Six inputs describe the equation, and the tool walks through the full solution rather than just handing back an answer.

How to use it

1. Enter the six coefficients a, b, c, d, e, f for the equation shown.
2. Press Calculate.
3. Read the worked steps down to the final solution set.

The equation it solves

The tool handles a general form of absolute value equation, written as a times the absolute value of b x plus c, plus d, equal to e x plus f. That looks busy, but it just means there is an absolute value expression on the left, scaled and shifted, set equal to a straight-line expression on the right. By letting you set all six coefficients, the tool covers a wide range of equations in one place, from simple ones like the absolute value of x minus 3 equals 5, up to versions with a variable on both sides. The first step it takes is to isolate the absolute value on its own.

The two-case method

The key to solving any absolute value equation is that an absolute value can be undone in two ways. Because the absolute value of something equals a value only if the thing inside is either that value or its negative, every absolute value equation splits into two ordinary equations. In one case the expression inside the bars is taken as positive, and in the other it is taken as negative, with a minus sign applied to the other side. The tool forms both of these cases, drops the absolute value bars, and is left with two plain linear equations, each of which it solves for x.

Checking each solution

Splitting into two cases can produce two candidate answers, but not every candidate is genuine. Each case is only valid over the range where its assumption holds, that is, where the expression inside the bars really is positive, or really is negative. So after solving, the tool checks each candidate against the condition for its case. A solution is kept only if it actually falls in the range that case assumed. This checking step is what stops absolute value equations from producing false answers, and it is why the tool reports the conditions alongside the working.

A worked example

Take the absolute value of x minus 3 equals 5. The positive case says x minus 3 equals 5, giving x equals 8. The negative case says x minus 3 equals negative 5, giving x equals negative 2. The dividing point is where x minus 3 is zero, at x equals 3. The candidate 8 sits above 3, matching its case, and negative 2 sits below 3, matching its case, so both are valid. The solution set is x equals 8 and x equals negative 2.

Two, one, or no solutions

Absolute value equations do not always have two answers. Often both cases give valid solutions, as in the example above, so there are two. Sometimes only one case survives the condition check, leaving a single solution. And sometimes neither candidate is valid, or the isolated absolute value is asked to equal something negative, which is impossible because an absolute value can never be negative. In that situation there is no solution at all, and the tool reports the empty set. To simply evaluate an absolute value rather than solve an equation, the absolute value calculator is the tool for that.

Questions people ask

What kind of equation does this solve?

A general absolute value equation, with an absolute value expression on one side set equal to a linear expression on the other.

Why are there two cases?

Because the thing inside the absolute value bars can be either positive or negative and still give the same absolute value, so the equation splits into two.

Why check the solutions?

Each case is valid only where its sign assumption holds. Checking each candidate against that condition removes false answers.

Can there be no solution?

Yes. If neither candidate is valid, or the absolute value is set equal to a negative amount, there is no solution and the tool reports the empty set.

What if I just want to evaluate an absolute value?

Then use the absolute value calculator, which returns the absolute value of a single number.

References

On absolute value equations. An absolute value equation is solved by splitting it into two cases, since the expression inside the bars may be positive or negative.

1. Eric W. Weisstein, "Absolute Value," from MathWorld, a Wolfram resource, on the absolute value, whose two-sided nature drives the two-case method.
2. "Linear Inequalities and Absolute Value Inequalities," Algebra and Trigonometry, OpenStax, on working with absolute value in equations and inequalities.