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Absolute Value Calculator
Find the absolute value of a number, or solve |ax + b| = c with both cases, one solution when c is 0, and no solution when c is negative.
Negatives, decimals, and zero are all fine
Formula
|n| is the distance of n from 0
Enter a number to see its absolute value, the distance from zero.
This absolute value calculator finds the absolute value of a number and solves equations written as |ax + b| = c. Type a number to see its distance from zero, or enter a, b, and c to see each case with those numbers substituted and the solutions listed. When c is negative, the page explains that there is no solution, because an absolute value cannot be negative. When c is zero, it shows the single solution of ax + b = 0.
A number's distance from zero, or the equation |ax + b| = c
Absolute value is the distance a number is from zero, so it is never negative. In the number mode, enter a value such as -7, 2.5, or 0 and the page shows the absolute value with a one-line reason. The absolute value of -7 is 7, because -7 is 7 units from zero.
The equation mode solves |ax + b| = c, as in |2x - 4| = 10. If c is positive, it writes the two cases ax + b = c and ax + b = -c, substitutes your numbers, and lists both solutions. If c is 0, there is one solution. If c is negative, there is no solution. The coefficient a cannot be 0, because then the equation has no x.
Frequently Asked Questions
What does absolute value mean?
Absolute value is how far a number is from zero on the number line. The bars in |n| mean that distance. It is never negative: the absolute value of -7 is 7, the absolute value of 7 is 7, and the absolute value of 0 is 0.
How do you solve |ax + b| = c?
Compare c with zero. If c is positive, solve the two linear equations ax + b = c and ax + b = -c. If c is zero, solve ax + b = 0, which gives the one solution x = -b / a. If c is negative, there is no solution. a must not be zero, or the equation has no x.
Why does a negative c have no solution?
An absolute value is a distance from zero, so it is always zero or positive. A negative number is not a distance. If c is negative, no value of x can make |ax + b| equal c. The equation |x| = -3 has no solution.
What are the solutions of |2x - 4| = 10?
Here a is 2, b is -4, and c is 10. The first case is 2x - 4 = 10, so 2x = 14 and x = 7. The second case is 2x - 4 = -10, so 2x = -6 and x = -3. Both check: the absolute value of 2(7) - 4 is 10, and the absolute value of 2(-3) - 4 is 10.