What Is Absolute Value? — Definition and Key Properties

This absolute value calculator handles everything related to absolute value: solving absolute value equations of the form |ax+b|=c, solving absolute value inequalities like |ax+b|<c or |ax+b|≥c with full interval notation, and graphing the absolute value function y=a|x−h|+k with the vertex, intercepts, and axis of symmetry all labeled.

Absolute value is defined as the distance a number is from zero on the number line. Because distance can never be negative, absolute value is always non-negative: |3|=3, |−3|=3, and |0|=0. Formally, |x| = x when x ≥ 0, and |x| = −x when x < 0.

Some students first encounter this concept under the name modulus — this is the same idea as absolute value, just a different name used more commonly in UK textbooks, engineering, and physics contexts. Whether you call it absolute value or modulus, the notation |x| and the underlying rules are identical.

The key properties of absolute value that make solving absolute value equations and solving absolute value inequalities possible are:

• Non-negativity: |x| ≥ 0 for every real number x
• Even function: |−x| = |x| — the graph of absolute value is symmetric about the y-axis
• Multiplicative property: |xy| = |x|·|y|
• Triangle inequality: |x+y| ≤ |x|+|y|
• Distance interpretation: |x−a| = the distance between x and a on the number line

Solving Absolute Value Equations — Step-by-Step

The fundamental approach to solving absolute value equations is recognising that |expression| = c creates two separate linear equations when c is positive, one equation when c equals zero, and no solution at all when c is negative. This absolute value calculator applies exactly this logic every time you solve an equation.

The Three Cases

• Case c > 0: Two solutions exist. Split into ax+b = c AND ax+b = −c, then solve each equation separately.
• Case c = 0: Exactly one solution exists, found by solving ax+b = 0.
• Case c < 0: No solution exists — absolute value can never equal a negative number.

Solving Absolute Value Inequalities — The AND vs OR Rule

The single most important conceptual distinction when solving absolute value inequalities is this: |ax+b| < c uses AND logic (a bounded interval — x is WITHIN a distance of the center), while |ax+b| > c uses OR logic (a union of rays — x is OUTSIDE a distance of the center).

Interval Notation and Number Lines for Absolute Value Solutions

Reading and writing interval notation correctly is a major source of confusion for students working through absolute value inequalities. Parentheses ( ) mean the endpoint is NOT included (open circle, strict inequality); brackets [ ] mean the endpoint IS included (closed/filled circle, non-strict inequality).

Every solution to an absolute value inequality can be written four equivalent ways: as a compound inequality, in interval notation, in set-builder notation, and as a number line diagram. Learning to convert fluently between all four representations of interval notation is essential.

Compound	Interval Notation	Set-Builder
−3 < x < 2	(−3, 2)	{x | −3<x<2}
x ≤ −1 or x ≥ 5	(−∞,−1]∪[5,+∞)	{x|x≤−1 or x≥5}
x = 4	[4, 4]	{4}
all reals	(−∞, +∞)	{x | x∈ℝ}

The interval notation calculator built into this tool draws the number line automatically for every absolute value inequality you solve, showing exactly where the open or closed circles belong — eliminating the most common source of wrong interval notation.

Graphing Absolute Value Functions — y = a|x − h| + k

Every absolute value function graphs as a V-shape. The vertex sits at the point (h, k) — this is the sharp corner where the two straight-line arms of the graph meet. The graph opens upward if a > 0 and downward if a < 0.

• a controls steepness and direction — larger |a| means steeper arms; a < 0 flips the graph upside down.
• h shifts the vertex horizontally — note that |x−h| shifts RIGHT by h when h > 0.
• k shifts the vertex vertically — up if k > 0, down if k < 0.

To find x-intercepts, set y = 0 and solve |x−h| = −k/a. To find the y-intercept, substitute x = 0 into the equation.

Special Cases — No Solution, One Solution, All Real Numbers

These edge cases are the most exam-tested part of absolute value equations and absolute value inequalities.

Absolute Value in Real Life — Distance Interpretation

The number line distance model makes abstract absolute value rules concrete. |x−3| < 5 asks "how far is x from 3? Less than 5" → x is between −2 and 8. |x−7| > 2 asks "is x more than 2 units from 7?" → x < 5 or x > 9.

Engineering tolerance is a classic real-world application: a manufactured part must be within 0.05mm of a target 10mm measurement, written as |measurement − 10| ≤ 0.05. Error bounds in scientific measurement follow the same pattern: |measured − actual| < tolerance.

Common Mistakes With Absolute Value

1. Forgetting to check if c < 0 before setting up cases — solving |2x+1| = −5 as two cases gives wrong answers; the correct answer is "no solution."
2. Swapping the AND/OR rule — using AND for > and OR for < is backwards. Remember: Less Than → beTween (AND), Greater Than → OR.
3. Forgetting to flip inequality signs when dividing a compound inequality by a negative number.
4. Thinking |x+3| = |x|+3 — absolute value does NOT distribute over addition. This is a very common algebra error.
5. Confusing the vertex shift direction — if the equation shows |x−h| and h > 0, the vertex shifts RIGHT, not left.

Worked Examples — All Three Tools

Frequently Asked Questions

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