Absolute Value Equation Calculator
Enter the coefficients of your absolute value equation in the form |ax + b| + c = d and this calculator solves it instantly. It finds both roots, shows the step-by-step split into positive and negative cases, tells you whether there are two solutions, one solution, or no solution, and plots the solutions on a number line. Switch to inequality mode to solve |ax + b| + c < d or > d as well.
Formula
Worked example
Solve |2x + 3| + 1 = 8. Step 1: subtract 1 from both sides to get |2x + 3| = 7. Step 2: split into 2x + 3 = 7 and 2x + 3 = -7. Step 3: solve each - from the first, 2x = 4 so x = 2; from the second, 2x = -10 so x = -5. Both solutions check out.
What is an absolute value equation?
An absolute value equation contains the absolute value operator |...|, which returns the non-negative magnitude of any real number. The equation |ax + b| = k has two solutions when k is positive, one solution when k is zero, and no real solutions when k is negative, because absolute values are always greater than or equal to zero. Understanding this three-case structure is the foundation of solving every absolute value problem.
How to solve |ax + b| + c = d step by step
The standard method has three steps. First, isolate the absolute value on one side: subtract c from both sides to get |ax + b| = d - c. Let k = d - c. If k < 0, stop - there is no solution. Second, split the equation into two linear cases: Case 1 is ax + b = k and Case 2 is ax + b = -k. Third, solve each linear equation for x by applying standard algebra. Case 1 gives x = (k - b) / a. Case 2 gives x = (-k - b) / a. Always verify both answers by substituting back into the original equation.
Solving absolute value inequalities
Absolute value inequalities follow a similar split strategy but produce interval solutions instead of discrete points. For |ax + b| < k (with k positive), the solution is a bounded interval: -k < ax + b < k, which simplifies to the open interval between the two boundary points. For |ax + b| > k, the solution is two unbounded rays: ax + b < -k or ax + b > k. The boundary points themselves are found by solving the corresponding equation |ax + b| = k, giving x values that separate the solution from the non-solution region.
Common mistakes and how to avoid them
The most frequent errors are: (1) Forgetting to isolate the absolute value first before splitting into cases - if c is non-zero, move it to the right side before splitting. (2) Forgetting the negative case - |ax + b| = k produces two equations, not one. (3) Failing to verify solutions - occasionally an algebraic manipulation produces a value that does not actually satisfy the original equation (an extraneous solution), so always substitute back. (4) Assuming every absolute value equation has two solutions - if k = 0 there is only one, and if k < 0 there are none.
Absolute value equation types and their solutions
| Condition | Number of solutions | Form of solutions | Example |
|---|---|---|---|
| k > 0 | 2 solutions | x = (k - b)/a and x = (-k - b)/a | |x - 3| = 5 gives x = 8, x = -2 |
| k = 0 | 1 solution | x = -b/a | |x - 3| = 0 gives x = 3 |
| k < 0 | No solution | None (|...| cannot be negative) | |x - 3| = -2 has no solution |
Summary of all cases for |ax + b| = k (where k = d - c after isolating the absolute value).
Frequently asked questions
Why does |ax + b| = k give two solutions?
The absolute value of a number equals k whenever the number itself is k or -k. So |ax + b| = k is really two equations in disguise: ax + b = k (the positive case) and ax + b = -k (the negative case). Each linear equation gives one value of x, so the original absolute value equation has two solutions when k is positive.
What does it mean when there is no solution?
If the right-hand side, after isolating the absolute value, is a negative number, the equation has no solution. An absolute value can never be negative (it is always zero or positive), so no value of x can satisfy |ax + b| = negative number. For example, |x + 5| = -3 has no solution.
How do I handle |ax + b| + c = d when c is not zero?
Subtract c from both sides first to isolate the absolute value: |ax + b| = d - c. Then proceed with the standard two-case split using k = d - c. If d - c is negative, there is no solution. This calculator does this isolation step automatically.
How does solving an absolute value inequality differ from an equation?
An equation |ax + b| = k gives discrete point solutions. An inequality gives an interval. For |ax + b| < k the solution is the open interval between the two boundary points (the values where equality holds). For |ax + b| > k the solution is the union of two rays extending outward from those boundary points. The boundary points are found by solving the equation first.
What if the coefficient a is negative?
The calculator handles negative a correctly. For example, |-2x + 6| = 8 gives Case 1: -2x + 6 = 8, so x = -1; and Case 2: -2x + 6 = -8, so x = 7. The formula x = (k - b) / a works for any non-zero a, positive or negative. The only restriction is a cannot be zero, because then there is no variable inside the absolute value.
How do I check whether my solution is correct?
Substitute each solution back into the original equation and verify that both sides are equal. For |2x + 3| = 7 with solution x = 2: |2(2) + 3| = |7| = 7, which matches. For x = -5: |2(-5) + 3| = |-7| = 7, which also matches. Both solutions are confirmed.