Absolute Value Calculator
Free absolute value calculator. Find the absolute value of any number or expression. Includes explanation, examples, and the absolute value formula.
Absolute Value Calculator
Comprehensive Guide: Absolute Value, Distance on Number Line & Modulus
Learn the piecewise algebraic definition of absolute value, geometric distance interpretation, solving absolute value equations, and inequalities.
1. Mathematical Definition & The Piecewise Function
The absolute value (or modulus) of a real number x, denoted symbolically as |x|, represents its non-negative magnitude regardless of direction or algebraic sign. Formally, absolute value is defined as a piecewise function:
|x| = −x (if x < 0)
Notice that when x is negative, −x is strictly positive: |−7| = −(−7) = +7. In analytic terms, absolute value is also equivalent to the principal square root of the square: |x| = √(x²).
2. Geometric Interpretation: Distance on the Real Number Line
Geometrically, |x| represents the physical distance between the point x and the origin 0 on the one-dimensional Euclidean number line. Because physical distance is inherently non-negative, absolute value can never evaluate to a negative number.
More broadly, the distance between any two real coordinates a and b is measured as:
The distance between −4 and +5 is |−4 − 5| = |−9| = 9 units.
3. Graduated Step-by-Step Worked Problems
Problem 1: Solve |2x − 5| = 9
Case 1 (Positive branch):
2x − 5 = 9 ⟹ 2x = 14 ⟹ x₁ = 7.
Case 2 (Negative branch):
2x − 5 = −9 ⟹ 2x = −4 ⟹ x₂ = −2.
Verification:
• For x = 7: |2(7) − 5| = |14 − 5| = |9| = 9 ✓
• For x = −2: |2(−2) − 5| = |−4 − 5| = |−9| = 9 ✓
Solutions: x ∈ {−2, 7}.
Problem 2: Solving an Absolute Value Inequality (|3x + 2| ≤ 11)
• Rewrite as a continuous compound inequality: −11 ≤ 3x + 2 ≤ 11.
• Subtract 2 from all three sides: −13 ≤ 3x ≤ 9.
• Divide by 3: −13/3 ≤ x ≤ 3 (or approx. −4.33 ≤ x ≤ 3.00).
Solution: The closed interval [−13/3, 3].
Problem 3: Equation with Two Absolute Values (|x − 3| = |2x + 1|)
Branch into two algebraic equations:
• Positive branch: x − 3 = 2x + 1 ⟹ x₁ = −4.
• Negative branch: x − 3 = −(2x + 1) = −2x − 1 ⟹ 3x = 2 ⟹ x₂ = 2/3.
Solutions: x ∈ {−4, 2/3}.
4. Real-World Applications in Engineering Tolerances & Physics
In manufacturing and quality control, technical specifications are defined using absolute value tolerance limits. If a piston diameter must measure 50 mm ± 0.05 mm, any valid manufactured part must satisfy the inequality |d − 50| ≤ 0.05.
5. Frequently Asked Questions (FAQ)
Can an absolute value equation have no solution?
Yes. If an equation states |2x + 1| = −5, there are zero solutions (empty set ∅), because absolute value can never evaluate to a negative number.
What does absolute value mean for complex numbers?
For a complex number z = a + bi, its absolute value (or modulus) is the Euclidean distance from the origin in the complex plane: |z| = √(a² + b²).
Is √(x²) always equal to x?
No, this is a classic mathematical trap! √(x²) = |x|. For example, √((−5)²) = √25 = +5, not −5. The square root symbol always denotes the non-negative principal root.
What is the Triangle Inequality for absolute values?
The Triangle Inequality states that for any real numbers x and y: |x + y| ≤ |x| + |y|. Equality holds if and only if both numbers share the same sign or at least one is zero.