Absolute Value Equation Calculator
Our free algebra calculator solves absolute value equation problems. Get worked examples, visual aids, and downloadable results.
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.
How do you verify solutions to an absolute value equation?
To verify solutions, substitute each value back into the original equation and confirm that both sides are equal. For example, if solving |2x - 1| = 5 gives x = 3 and x = -2, check x = 3: |2(3) - 1| = |5| = 5, which equals the right side. Check x = -2: |2(-2) - 1| = |-5| = 5, which also equals the right side. Verification is especially important when dealing with more complex equations that involve squaring or other operations that might introduce extraneous solutions not valid in the original equation context.
Can absolute value equations contain variables on both sides?
Yes, absolute value equations can have variables on both sides, such as |2x + 1| = |x - 3|. To solve these, you consider two cases: either the expressions inside both absolute values are equal (2x + 1 = x - 3), or they are negatives of each other (2x + 1 = -(x - 3)). This approach works because two numbers have the same absolute value if and only if they are equal or they are opposites. These types of equations commonly appear in distance problems where you need to find points equidistant from two reference locations on the number line.