Absolute Value Calculator
Evaluate absolute values, solve absolute value equations and inequalities, measure distance between two points, and graph V-shaped absolute value functions — each with its own visual and full step-by-step working.
Background
The absolute value of a number is its distance from zero on the number line — always zero or positive, no matter which direction the number sits. That single idea shows up everywhere: solving |x|=c, describing a tolerance range like |x−a|<c, measuring how far apart two points are, and drawing the sharp-cornered V-shaped graphs that absolute value functions always produce.
How to use this calculator
- Evaluate lets you type any numeric expression and see its absolute value, with the distance-from-zero shown on a number line.
- Solve Equation handles |ax+b| = c by splitting it into its two cases and marking both solutions.
- Solve Inequality handles |ax+b| < c, ≤ c, > c, or ≥ c, shading the exact solution region.
- Distance finds how far apart two numbers are on a number line — the same idea as absolute value, applied to two points instead of one — and also reports the midpoint between them.
- Graph It draws the V-shaped graph of y = a|x−h|+k and marks the vertex, axis of symmetry, and both intercepts.
- Click Calculate to see the visual and the full step-by-step working for whichever mode you picked.
How absolute value works
Absolute value is distance from zero, so it is always zero or positive — |x|=x when x is already zero or positive, and |x|=−x (which flips a negative into a positive) when x is negative.
Absolute value equations split into two linear cases because the expression inside the bars could be positive or negative before the bars are applied — setting it equal to both +c and −c covers both possibilities.
|expr| < c is an "and" condition — the expression must stay close to zero — which produces one connected interval. |expr| > c is an "or" condition — the expression must be far from zero in either direction — which produces two separate unbounded rays.
The distance between two numbers, |a−b|, is just absolute value applied to a difference instead of a single value — and it's symmetric, since |a−b| = |b−a|.
The graph of y = a|x−h|+k is always a V-shape with a sharp corner at the vertex (h, k) — that corner is essential and can't be smoothed away by shifting or scaling.
Formula & Equations Used
Definition: |x| = x if x ≥ 0, and |x| = −x if x < 0.
Alternate form: |x| = √(x²)
Equation: |ax+b| = c → x = (c−b)/a or x = (−c−b)/a, valid only when c ≥ 0.
Less-than inequality: |ax+b| < c → center = −b/a, radius = c/|a|, solution is one interval around the center.
Greater-than inequality: |ax+b| > c → same center and radius, but the solution is everything outside that interval.
Distance: d = |a − b| = |b − a|
Midpoint: m = (a + b) / 2 — exactly halfway between the two points.
X-intercepts of the graph: set y = 0 in y = a|x−h|+k and solve — the same case-split as the Equation mode above.
Example Problems & Step-by-Step Solutions
These cover cases the Quick Examples chips above don't already demonstrate.
Example 1 — Evaluate a real-world drop
A submarine dives to an elevation of −187 m relative to sea level. How far below the surface is it?
Step: Depth below surface = |−187|.
Result: 187 m below the surface.
Example 2 — An equation with a real constraint
A machined part must satisfy |2x+5| = 13. Find every valid x.
Step: 2x+5=13 → x=4. 2x+5=−13 → x=−9.
Result: x = 4 or x = −9.
Example 3 — An inequality with units
A server room's temperature T (°F) must satisfy |T−72| ≤ 4 to stay safe. Find the safe range.
Step: Center = 72, radius = 4, so 72−4 ≤ T ≤ 72+4.
Result: T is between 68°F and 76°F, inclusive.
Example 4 — Reading a graph in context
A tent's cross-section is modeled by y = −0.5|x| + 6 (feet), x measured from the center pole. Find the peak height and the height 4 ft from center.
Step: Vertex (peak) is at (0, 6). At x=4: y = −0.5(4)+6 = 4.
Result: Peak height 6 ft; height at 4 ft from center is 4 ft.
Frequently Asked Questions
Why is absolute value never negative?
Because it measures distance, and distance can't be negative — even when the number itself is negative, its absolute value flips it to the positive distance from zero.
Is absolute value just about erasing a minus sign?
That works for a single negative number, but for an expression like |x−5| there's no minus sign to erase until you know whether x−5 turns out positive or negative — which is exactly why equations and inequalities need case-splitting instead of just deleting a sign.
Why do absolute value equations split into two cases?
|expression| = c means the expression sits exactly c units from zero, and there are always two spots that far from zero: +c and −c. Setting the expression equal to each one covers both possibilities.
Why do "less than" and "greater than" solutions look so different?
|expr| < c means the expression must stay close to zero on both sides, giving one connected interval. |expr| > c means the expression must be far from zero in either direction, giving two separate rays — an "or" instead of an "and."
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