What is absolute value?

Absolute value is the distance of a number from zero on the number line, regardless of direction. It is always zero or positive. The absolute value of -7 is 7, and the absolute value of 7 is also 7. Both are 7 units away from zero. In math notation, absolute value is written with vertical bars: |x|. For example, |-7| = 7 and |7| = 7. The formal definition is: |x| = x if x >= 0, and |x| = -x if x < 0.

How do you calculate absolute value?

To calculate the absolute value of a number, simply remove the negative sign if there is one. If the number is already positive or zero, it stays the same. Examples: |5| = 5, |-3| = 3, |0| = 0, |-2.5| = 2.5. For fractions, take the absolute value of the entire fraction: |-3/4| = 3/4. For expressions like |3x - 7|, you need to know the value of x first, then evaluate the expression inside, then take the absolute value of the result.

How do you solve absolute value equations?

To solve |ax + b| = c, split the equation into two cases. Case 1: ax + b = c (the expression equals the positive value). Case 2: ax + b = -c (the expression equals the negative value). Solve each case separately for x. Important: if c is negative, there is no solution because absolute value cannot equal a negative number. If c = 0, there is exactly one solution. Our calculator shows these steps automatically.

How do you solve absolute value inequalities?

For |ax + b| c (greater than), split into two separate inequalities: ax + b c. The solution is two rays going in opposite directions. Remember: if dividing by a negative number, flip the inequality sign.

What is the absolute value of a fraction?

The absolute value of a fraction works the same as any number - take the distance from zero. So |-3/4| = 3/4, |5/8| = 5/8, and |-7/2| = 7/2 = 3.5. You can also think of it as taking the absolute value of the numerator and denominator separately: |a/b| = |a|/|b|. Our calculator supports fraction input using the slash notation (e.g., type -3/4).

What is the difference between absolute value and modulus?

For real numbers, absolute value and modulus mean the same thing. The term 'modulus' is more commonly used in some countries and in the context of complex numbers. For a complex number a + bi, the modulus is sqrt(a squared + b squared), which gives the distance from the origin in the complex plane. For real numbers, both terms refer to the distance from zero on the number line.

Can absolute value be negative?

No. Absolute value is defined as the distance from zero, and distance is always zero or positive. The absolute value of any real number is always greater than or equal to zero. This is the non-negativity property: |x| >= 0 for all x. This property is why |ax + b| = c has no solution when c < 0, and why |ax + b| < c has no solution when c < 0.

What are the properties of absolute value?

The key properties are: (1) Non-negativity: |a| >= 0 always. (2) |a| = 0 if and only if a = 0. (3) |ab| = |a| times |b| (multiplicativity). (4) |a + b| <= |a| + |b| (triangle inequality). (5) |-a| = |a| (symmetry). (6) |a/b| = |a|/|b| when b is not zero. These properties are fundamental in algebra, calculus, and real analysis.

Free Absolute Value Calculator

Absolute Value Calculator

Q: How do you calculate absolute value?

To calculate the absolute value of a number, simply remove the negative sign if there is one. If the number is already positive or zero, it stays the same. Examples: |5| = 5, |-3| = 3, |0| = 0, |-2.5| = 2.5. For fractions, take the absolute value of the entire fraction: |-3/4| = 3/4. For expressions like |3x - 7|, you need to know the value of x first, then evaluate the expression inside, then take the absolute value of the result.

Q: How do you solve absolute value equations?

To solve |ax + b| = c, split the equation into two cases. Case 1: ax + b = c (the expression equals the positive value). Case 2: ax + b = -c (the expression equals the negative value). Solve each case separately for x. Important: if c is negative, there is no solution because absolute value cannot equal a negative number. If c = 0, there is exactly one solution. Our calculator shows these steps automatically.

Q: How do you solve absolute value inequalities?

For |ax + b| c (greater than), split into two separate inequalities: ax + b c. The solution is two rays going in opposite directions. Remember: if dividing by a negative number, flip the inequality sign.

Q: What is the absolute value of a fraction?

The absolute value of a fraction works the same as any number - take the distance from zero. So |-3/4| = 3/4, |5/8| = 5/8, and |-7/2| = 7/2 = 3.5. You can also think of it as taking the absolute value of the numerator and denominator separately: |a/b| = |a|/|b|. Our calculator supports fraction input using the slash notation (e.g., type -3/4).

Q: What is the difference between absolute value and modulus?

For real numbers, absolute value and modulus mean the same thing. The term 'modulus' is more commonly used in some countries and in the context of complex numbers. For a complex number a + bi, the modulus is sqrt(a squared + b squared), which gives the distance from the origin in the complex plane. For real numbers, both terms refer to the distance from zero on the number line.

Q: Can absolute value be negative?

No. Absolute value is defined as the distance from zero, and distance is always zero or positive. The absolute value of any real number is always greater than or equal to zero. This is the non-negativity property: |x| >= 0 for all x. This property is why |ax + b| = c has no solution when c < 0, and why |ax + b| < c has no solution when c < 0.

Q: What are the properties of absolute value?

The key properties are: (1) Non-negativity: |a| >= 0 always. (2) |a| = 0 if and only if a = 0. (3) |ab| = |a| times |b| (multiplicativity). (4) |a + b| <= |a| + |b| (triangle inequality). (5) |-a| = |a| (symmetry). (6) |a/b| = |a|/|b| when b is not zero. These properties are fundamental in algebra, calculus, and real analysis.

Find the absolute value of any number, solve |ax + b| = c equations, and work through |ax + b| < c inequalities - all with step-by-step solutions you can follow and export.

Calculator Mode

Find Absolute Value

Enter a number (supports decimals and fractions like -2/3)

Quick examples:

Result

Enter a value to calculate

Properties of |x|

Non-negativity|a| ≥ 0

The absolute value is always zero or positive

Positive-definiteness|a| = 0 ⟺ a = 0

Absolute value is zero only when the number itself is zero

Multiplicativity|a × b| = |a| × |b|

The absolute value of a product equals the product of absolute values

Triangle inequality|a + b| ≤ |a| + |b|

The absolute value of a sum is at most the sum of absolute values

Symmetry|-a| = |a|

A number and its negative have the same absolute value

Division|a/b| = |a|/|b|

The absolute value of a quotient equals the quotient of absolute values

Common Absolute Values

Quick reference for frequently looked-up absolute values.

|-1|1

|0|0

|-5|5

|3.14|3.14

|-100|100

|-2/3|2/3

|-0.5|0.5

|7|7

How to Use the Absolute Value Calculator

Three modes, one tool. Pick what you need and get answers fast.

Basic Mode - Find |x|

Type any number - whole, decimal, or fraction (like -2/3) - and the calculator instantly shows the absolute value. A number line below the input marks where the original number sits and how far it is from zero. Tap any quick example to see it in action.

Equation Mode - Solve |ax + b| = c

Enter the coefficients a, b, and the constant c. The calculator splits the equation into two cases, works through each one, and hands you both solutions with every algebraic step written out. If there is no solution (c is negative) or only one solution (c is zero), it explains why.

Inequality Mode - Solve |ax + b| < c or > c

Enter your coefficients, pick the inequality direction (less than, less than or equal, greater than, greater than or equal), and enter c. The tool outputs the solution in both inequality notation and interval notation, with a full step-by-step breakdown. Share or export the result for homework or study notes.

Why Use This Calculator

Step-by-Step Solutions

Every equation and inequality is broken down into individual algebraic steps. You see exactly how each answer is reached, making it a study tool, not just an answer machine.

Three Modes in One

Basic absolute value, equation solving, and inequality solving are all here. Switch between modes without leaving the page or opening a different calculator.

Number Line Visual

The basic mode plots the original number and its absolute value on a number line so you can see the distance-from-zero concept instead of just reading about it.

Fraction Support

Type -3/4 directly and get the absolute value as a fraction. No need to convert to decimals first. The calculator understands slash notation natively.

Interval Notation

Inequality solutions come in both algebraic form and interval notation. Helpful for students who need to write answers in a specific format for class or standardized tests.

Share and Export

Copy your result to clipboard for messaging apps or export a full text file with the problem, solution, and all steps. Useful for homework submissions or tutoring sessions.

The Complete Guide to Absolute Value

What Absolute Value Actually Means

Strip away the textbook jargon and absolute value is just distance. Stand at zero on a number line: |-7| asks "how many steps to get to -7?" The answer is 7. |7| asks the same question for the positive side and gets the same answer. Direction does not matter, only magnitude. That single idea underpins everything else on this page.

Formally, |x| is defined as a piecewise function: it equals x when x is non-negative and equals -x when x is negative. Writing -x does not mean "make it negative" - it means "flip the sign." Flipping a negative gives you a positive, which is exactly the distance we want. The notation with vertical bars was introduced by Karl Weierstrass in 1841 and has been standard in mathematics ever since.

Solving Absolute Value Equations

An equation like |2x - 5| = 9 asks: "When is the expression 2x - 5 exactly 9 units away from zero?" That happens in two scenarios. Either 2x - 5 itself equals 9, or 2x - 5 equals -9. Solving the first gives x = 7, solving the second gives x = -2. Both satisfy the original equation, and plugging them back in confirms it.

The trap to watch for is a negative constant on the right side. |anything| = -3 has no solution because absolute value cannot produce a negative output. Students lose easy marks on exams by plowing through the algebra without checking this first. Another subtlety: |expression| = 0 means the expression inside is exactly zero, so there is only one solution, not two.

More complex equations might have absolute value on both sides, like |x - 1| = |2x + 3|. The strategy is similar - you consider four combinations of signs - but two of them will be duplicates, leaving two cases to solve. Our calculator handles the standard |ax + b| = c form. For double-absolute-value equations, apply the same split-into-cases logic by hand.

Working Through Absolute Value Inequalities

Inequalities with absolute value split into two flavors depending on the direction of the comparison. "Less than" problems trap the variable in a range: |x - 4| < 3 means x sits within 3 units of 4, giving 1 < x < 7. In interval notation that is (1, 7). Picture a number line: you are looking at the neighborhood around 4 that extends 3 units in each direction.

"Greater than" problems push the variable away from the center: |x - 4| > 3 means x is more than 3 units from 4, so x < 1 or x > 7. In interval notation, (-infinity, 1) union (7, infinity). The solution is two separate pieces of the number line instead of one connected interval.

A useful shortcut to remember: "less thAND" (compound inequality, one interval) versus "greatOR" (two separate inequalities joined by OR). This mnemonic has saved countless students on timed tests. Our calculator produces both forms of the answer plus every algebraic step so you can follow the reasoning.

Properties You Should Know

Six properties of absolute value come up repeatedly in algebra and beyond. Non-negativity (|a| is always at least 0) and positive-definiteness (|a| = 0 only when a = 0) are the foundation. Multiplicativity says |ab| = |a| times |b|, which extends naturally to division: |a/b| = |a|/|b|. Symmetry tells us |-a| = |a| - flipping the sign does not change the absolute value.

The triangle inequality, |a + b| <= |a| + |b|, is the workhorse property. It shows up in proofs about metric spaces, convergence of sequences, error bounds, and dozens of other places in mathematics. It essentially says that a detour can never be shorter than the direct route. If you remember one property beyond the definition, make it this one.

Absolute Value in Other Contexts

Beyond basic algebra, absolute value appears in calculus (limits, continuity, and the epsilon-delta definition), linear algebra (norms of vectors extend the idea of absolute value to higher dimensions), complex analysis (the modulus |a + bi| = sqrt(a squared + b squared)), and statistics (mean absolute deviation, mean absolute error).

In programming, the abs() function or Math.abs() is one of the most called utility functions. It shows up in distance calculations, sorting algorithms, error checking, and game physics. Financial applications use it to compute profit or loss magnitude regardless of direction. Signal processing relies on it when measuring amplitude without caring about phase.

Understanding absolute value at the conceptual level - distance from a reference point - makes all of these applications intuitive. Whether you are solving a homework problem or writing production code, the idea is the same: strip the sign and keep the magnitude.