Absolute Value Calculator
Absolute value answers a simple question: how far is this number from zero, ignoring direction? A temperature drop of 7.5 degrees and a rise of 7.5 degrees are opposite events with the same size, and absolute value is the tool that captures the size. That is why |-7.5| and |7.5| are both 7.5.
The Absolute Value Calculator below works in two modes, the same two modes as the original tool. Evaluate mode finds |x| for any real number and draws the point and its absolute value on a number line. Solve mode handles equations of the form |ax + b| = c with full step-by-step working, including the honest edge cases: two solutions when c is positive, one solution when c is zero, and no solution at all when c is negative.
Quick-pick chips under the input evaluate the values students ask about most, -15, -7.5, 0, 7.5, and 42, so you can watch the pattern before typing anything: negatives flip to their positive twin, zero stays zero, and positives pass through unchanged.
Quick Answer
Key takeaways
- |x| is distance from zero: |x| = x when x is zero or positive, and |x| = -x when x is negative. The result is never negative.
- To solve |ax + b| = c, split it into ax + b = c and ax + b = -c. Each branch is an ordinary linear equation.
- The value of c decides the solution count before you do any algebra: c positive gives two solutions, c = 0 gives one, c negative gives none.
- Every solution gets checked by substitution. The tool prints the check with the result, which is the habit that catches sign errors in homework.
How the absolute value is computed
The tool applies the piecewise definition directly, the same definition set out in the Khan Academy introduction to absolute value: if x is zero or positive, |x| = x; if x is negative, |x| = -x, and the two minus signs cancel to leave a positive result. Alongside the value, it reports how the input is classified (positive, negative, or zero), the distance from zero, and the opposite number, and it draws both points on the number line so you can see that x and -x sit the same distance from zero in opposite directions.
There is a second definition worth knowing, because it explains several facts at once: |x| is the square root of x squared. Squaring erases the sign, and the square root returns the positive root, so |-7.5| = sqrt(56.25) = 7.5. Root tools such as the fourth root calculator work the same positive-root convention from the other direction.
| x | |x| | Opposite of x | Distance from zero |
|---|---|---|---|
| -15 | 15 | 15 | 15 units |
| -7.5 | 7.5 | 7.5 | 7.5 units |
| 0 | 0 | 0 | 0 units |
| 7.5 | 7.5 | -7.5 | 7.5 units |
| 42 | 42 | -42 | 42 units |
Notice the one asymmetry in the table: the absolute value and the opposite coincide only when x is negative or zero. For positive x, the opposite is negative while the absolute value stays positive. That distinction is the single most common source of wrong answers in this topic, and it is why the calculator prints both numbers side by side.
Solving |ax + b| = c, step by step
Read c first. If c is negative, stop: an absolute value can never equal a negative number, so there is no solution. If c is zero, the expression inside the bars must be zero, and there is exactly one solution, x = -b/a.
Split into two cases. When c is positive, |u| = c means u = c or u = -c, because both c and -c sit exactly c units from zero. Replace u with ax + b to get the two branch equations.
Solve each branch. Each branch is a plain linear equation: subtract b, divide by a. The branch-solving mechanics are the same isolating-x routine used by the linear function equation calculator, one inverse operation at a time.
Substitute back and check. Put each candidate into the original equation. Both must return exactly c inside the bars. The tool prints this check automatically; on paper, it takes ten seconds and is worth it every time.
Worked example, verified against the tool
Solve |-2x + 6| = 10, the pre-filled equation in Solve mode. Here a = -2, b = 6, c = 10:
| Step | Working | Result |
|---|---|---|
| Split | -2x + 6 = 10 or -2x + 6 = -10 | Two branch equations |
| First branch | -2x = 4, so x = 4 / -2 | x = -2 |
| Second branch | -2x = -16, so x = -16 / -2 | x = 8 |
| Check x = -2 | |-2(-2) + 6| = |10| | 10, correct |
| Check x = 8 | |-2(8) + 6| = |-10| | 10, correct |
The tool returns exactly these two solutions with the same three steps and both checks. Run the companion cases to see the pattern hold: set c to 0 and the same a and b give the single solution x = 3; set c to -4 and the tool stops at step 1 with no solution, before any splitting happens.
Where does this show up outside algebra class? Anywhere direction does not matter but size does. Error margins and tolerances in engineering are absolute values (a part is within 0.02 mm of spec, either way). Deviations from a mean in statistics are averaged as absolute values in the mean absolute deviation, a close cousin of the percentage averaging done by the average of percentages calculator. Price moves in finance are quoted as absolute changes when the direction is reported separately. The pattern is always the same: strip the sign, keep the magnitude.
What competing calculators miss
We reviewed the top-ranking absolute value tools before rebuilding this page. The big course-platform calculators are genuinely broad, and the small ones are fast. The gaps sit between those poles.
Winning move 1: both modes, one tool, no mode confusion. Several ranking tools evaluate |x| only, or solve equations only, so a student with both question types bounces between pages with different conventions. This page keeps the original tool’s two tabs: evaluation with a number line, and equation solving with steps, sharing one result box and one check habit.
Winning move 2: the edge cases are answered, not errored. Thin solvers silently return nonsense or a blank for c negative, c = 0, or a = 0. This tool names each case and explains it: no solution because an absolute value cannot be negative, one solution because only zero has absolute value zero, and the a = 0 collapse where the equation is true for every x or for none.
Winning move 3: a real number line, drawn from your numbers. Most text-only solvers print the answer and a generic diagram. The evaluate mode draws the number line for your input, marking x, zero, and |x| at their true positions, with the scale widening automatically for large inputs such as the 42 chip. Seeing -7.5 and 7.5 land on the same marked distance is the lesson; the picture teaches it faster than the definition.
Winning move 4: the check is part of the answer. Competing step-by-step tools often stop at the two candidate solutions. This one substitutes both back into the original equation and prints the verification, because in this topic the extraneous-looking errors are almost always sign slips that a ten-second check exposes.
Frequently asked questions
It is the distance of the number from zero on the number line, ignoring direction. Formally, |x| = x when x is zero or positive, and |x| = -x when x is negative. So |-9| = 9 and |9| = 9, and |-7.5| = 7.5. The result is always zero or positive, never negative.
Never. An absolute value measures distance, and distance cannot be negative. That single fact decides whole problems before any algebra: |x| = -3 has no solution, while |x| = 3 has two solutions, x = 3 and x = -3, because both sit 3 units from zero.
Rewrite it as two equations, ax + b = c and ax + b = -c, solve each for x, then substitute both answers back into the original equation to confirm. For |-2x + 6| = 10, the branches give x = -2 and x = 8, and both check: |-2(-2) + 6| = 10 and |-2(8) + 6| = 10. This tool shows every step and both checks.
Because both 5 and -5 sit exactly 5 units from zero. The absolute value erases the sign, so the equation cannot tell the two apart and both qualify. In general, |ax + b| = c with c positive has two solutions, which is why the solver always splits into two branch equations.
You get a single solution, x = -b/a. Only the number zero has an absolute value of zero, so the expression inside the bars must equal zero, and one linear equation gives one answer. With a = -2 and b = 6, that is x = 3. This is the boundary case between two solutions (c positive) and none (c negative).
The opposite of x is -x. For any nonzero x, exactly one of x and -x is positive, and |x| equals whichever one is positive. That is why |-7.5| and |7.5| are both 7.5. The calculator prints the opposite alongside the result because confusing the two is the most common error in this topic: the opposite of 7.5 is -7.5, but its absolute value is 7.5.
Yes. Enter any real number, including decimals and large values, and the tool evaluates |x| exactly, widening the number line scale when the input falls outside the default range. The equation solver accepts decimal coefficients for a, b, and c as well, and reports solutions trimmed of floating-point noise.
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References
- Khan Academy, “Intro to Absolute Value” (Algebra basics) – the distance-from-zero definition, the piecewise rule, and number line examples used in classrooms worldwide. Khan Academy: Intro to Absolute Value. Accessed October 2026.
- Encyclopaedia Britannica, “Absolute Value” – the reference definition of absolute value as the magnitude of a real number regardless of sign, with the distance interpretation. Encyclopaedia Britannica: Absolute Value. Accessed October 2026.
- Absolute value, reference overview – the piecewise definition, distance interpretation, and properties including the triangle inequality. Absolute Value: Definition and Properties. Accessed October 2026.
- Purplemath (Elizabeth Stapel), “Absolute Value: Concepts and Solving” – the widely used algebra lessons on the piecewise definition, the two-case split for equations, and the no-solution case for negative right-hand sides. Purplemath: Absolute Value Lessons. Accessed October 2026.
- National Council of Teachers of Mathematics, “Principles and Standards for School Mathematics” – the standards framework for how absolute value and distance on the number line are taught in school algebra. NCTM: Principles and Standards for School Mathematics. Accessed October 2026.