Absolute Value Equation Calculator
Our free algebra calculator solves absolute value equation problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
|ax + b| = c
Where a is the coefficient of x, b is the constant term inside the absolute value, and c is the value the absolute value expression equals. When c > 0, split into two cases: ax + b = c and ax + b = -c. When c = 0, one solution. When c < 0, no solution.
Worked Examples
Example 1: Solving |2x - 3| = 7
Problem:Find all values of x satisfying |2x - 3| = 7.
Solution:Case 1: 2x - 3 = 7 => 2x = 10 => x = 5\nCase 2: 2x - 3 = -7 => 2x = -4 => x = -2\nVerify: |2(5) - 3| = |7| = 7 and |2(-2) - 3| = |-7| = 7. Both check out.
Result:x = -2 and x = 5
Example 2: Solving |x + 4| = 0
Problem:Find all values of x satisfying |x + 4| = 0.
Solution:Since absolute value equals zero only when the inside expression equals zero:\nx + 4 = 0 => x = -4\nVerify: |(-4) + 4| = |0| = 0. Correct.
Result:x = -4 (unique solution)
Frequently Asked Questions
What is an absolute value equation and how do you solve one?
An absolute value equation is an equation that contains an expression inside absolute value bars, such as |ax + b| = c. The absolute value of a number represents its distance from zero on the number line, so it is always non-negative. To solve an absolute value equation, you split it into two separate linear equations: one where the expression inside equals the positive value, and one where it equals the negative value. For example, |2x + 3| = 7 becomes 2x + 3 = 7 and 2x + 3 = -7, yielding x = 2 and x = -5 as the two solutions.
When does an absolute value equation have no solution?
An absolute value equation has no solution when the expression is set equal to a negative number. Since the absolute value function always returns a non-negative result (zero or positive), it is mathematically impossible for |ax + b| to equal any negative number. For instance, the equation |3x - 4| = -2 has no solution because no matter what value of x you substitute, the left side will always be zero or positive. Recognizing this condition early saves time and prevents unnecessary algebraic manipulation in problem-solving scenarios.
How many solutions can an absolute value equation have?
A standard absolute value equation of the form |ax + b| = c can have zero, one, or two solutions depending on the value of c. When c is negative, there are no solutions because absolute values cannot be negative. When c equals zero, there is exactly one solution because the expression inside the absolute value bars must itself equal zero. When c is positive, there are exactly two solutions corresponding to the positive and negative cases. More complex equations involving multiple absolute value terms or higher-degree polynomials may have additional solutions requiring piecewise analysis.
What is the difference between absolute value equations and absolute value inequalities?
Absolute value equations like |ax + b| = c produce discrete point solutions, while absolute value inequalities like |ax + b| < c or |ax + b| > c produce interval solutions on the number line. For a less-than inequality, the solution is a bounded interval between two values. For a greater-than inequality, the solution consists of two unbounded rays extending outward. Both rely on the same fundamental principle of splitting into two cases, but inequalities require careful attention to the direction of inequality signs when removing the absolute value bars.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy