Introduction
Absolute value means distance from zero. It is never negative. So |5| is 5, and |−5| is also 5. Both numbers sit 5 steps away from zero on a number line.
This absolute value calculator handles five kinds of problems:
- Evaluate: find the absolute value of any expression, like |3 − 10|.
- Solve Equation: solve problems like |2x − 3| = 7 and get both answers.
- Solve Inequality: solve |x − 3| < 5 and see the answer in interval, inequality, and set-builder form.
- Distance: find how far apart two numbers are, plus the midpoint.
- Graph It: draw y = a|x − h| + k and find the vertex, axis of symmetry, and intercepts.
Type your numbers and press Calculate. You get the answer, a step-by-step solution, and a picture on a number line or graph. Use the Examples button to load a sample problem and see how it works.
How to use our Absolute Value Calculator
Pick a mode, type your numbers or expression, and the calculator shows the answer with a number line or graph plus step-by-step work.
Mode buttons: Choose Evaluate, Solve Equation, Solve Inequality, Distance, or Graph It at the top. The input boxes change to match your choice.
Expression (Evaluate mode): Type what goes inside the bars, like 3 - 10 or -2*(4-9). You can use +, −, ×, ÷, ^, and parentheses, or tap the symbol keys.
a, the coefficient of x (Equation and Inequality modes): Enter the number multiplied by x inside the bars. For |x − 3|, a is 1.
b, the constant inside the bars: Enter the number added or subtracted next to x. For |2x − 3|, b is −3.
c, the right-hand side: Enter the number on the other side of the equals or inequality sign.
Comparison operator (Inequality mode): Click <, ≤, >, or ≥ to set the sign in your inequality.
Point 1 (a) and Point 2 (b) (Distance mode): Enter the two numbers you want the distance between. Negatives are fine.
Unit label (Distance mode): Optional. Type a unit like meters or °C, and it is added to the distance and midpoint.
a, the stretch and direction (Graph It mode): Enter the number in front of the bars. Positive opens the V up; negative opens it down.
h, the horizontal shift: Enter how far the V moves left or right. This is the x-value of the vertex.
k, the vertical shift: Enter how far the V moves up or down. This is the y-value of the vertex.
Show visual and Show steps: Check these boxes to see the number line or graph and the full solution steps.
Calculate, Reset, and Copy Result: Click Calculate to solve, Reset to go back to the starting values, and Copy Result to save the answer.
What Is Absolute Value?
The absolute value of a number is how far that number sits from zero on a number line. Distance is never negative, so an absolute value is never negative. We write it with two straight bars, like |n|. For example, |7| = 7 and |−7| = 7. Both numbers are 7 steps away from zero, just in opposite directions.
The Rule in Plain Words
- If the number is positive, it stays the same: |5| = 5
- If the number is negative, drop the minus sign: |−5| = 5
- If the number is zero, the answer is zero: |0| = 0
When there is math inside the bars, always solve inside first, then take the absolute value. So |3 − 10| = |−7| = 7.
Distance Between Two Numbers
Absolute value is the easy way to find the gap between two points. The distance from a to b is |a − b|. The order does not matter, because |a − b| = |b − a|. For example, the distance from 5 to −3 is |5 − (−3)| = |8| = 8.
Absolute Value Equations
An equation like |ax + b| = c usually has two answers, because two different numbers can be the same distance from zero. You split it into two cases: ax + b = c and ax + b = −c. Two special cases to watch for:
- If c is negative, there is no solution. An absolute value can never be negative.
- If c is zero, there is only one answer, since the inside must equal zero.
Absolute Value Inequalities
Think of these as questions about distance:
- |u| < c means "closer than c to zero." The answer is one interval between two numbers: −c < u < c.
- |u| > c means "farther than c from zero." The answer is two separate pieces: u < −c or u > c.
Use ≤ and ≥ when the endpoints count. On a number line, a filled dot means the endpoint is included and a hollow dot means it is not.
Graphing y = a|x − h| + k
The graph of an absolute value function is a V shape. The point of the V is called the vertex, and it sits at (h, k). The vertical line x = h is the axis of symmetry, so both sides of the V mirror each other. If a is positive the V opens up; if a is negative it opens down. A bigger |a| makes the V narrower, and a smaller |a| makes it wider.
Where People Use It
Absolute value shows up any time size matters more than direction. Factories use it to check part sizes against a tolerance, like |length − 10| ≤ 0.5. Engineers use it for voltage limits, weather reports use it for temperature swings, and maps use it for elevation gaps between two places. It also shows how far a measurement is off from a target, which is called error.