Absolute Value
Absolute value measures how far a number is from zero on the number line, regardless of direction. It is a foundational concept in algebra that connects to equations, inequalities, graphing, and real-world distance applications.
This guide covers key definitions, properties, solving absolute value equations and inequalities, graphing V-shaped functions, distance applications, worked examples, common mistakes, and a practice quiz.
1Introduction
Imagine you are standing at point 0 on a number line. If you walk 5 units to the right, you are at point 5. If you walk 5 units to the left, you are at point -5. In both cases, the distance you walked is 5 units. This idea of distance, regardless of direction, is the core concept behind absolute value.
Absolute value is a fundamental concept in mathematics that helps us quantify "how far" a number is from zero, without considering its sign (positive or negative). It appears throughout algebra, geometry, calculus, and countless real-world applications.
If you owe $50, your bank balance might show -$50. But the amount you owe is $50 — the absolute value. Whether you gain or lose $50, the magnitude of the change is the same.
Real-World Uses
Measurement & Tolerances
A temperature of 20 +/- 2 degrees C means the actual temperature T satisfies |T - 20| ≤ 2.
Distance Calculations
The distance between two points on a map or number line: |a - b| is always positive.
Computer Science
Algorithms use absolute value for calculating differences, error margins, and deviations.
Physics
Speed is the absolute value (magnitude) of velocity. Force magnitudes ignore direction.
2Key Definitions
Absolute Value
The distance of a number from zero on the number line. Always non-negative. Notation: |x|.
Piecewise Definition
|x| = x if x ≥ 0, and |x| = -x if x < 0. The negative sign makes a negative number positive.
Magnitude
Synonym for absolute value. The "size" of a quantity without regard to sign or direction.
Distance
The distance between a and b is |a - b| or |b - a|. Always non-negative.
Non-negative
A value that is zero or positive. Absolute value is always non-negative: |x| ≥ 0.
Formal (Piecewise) Definition
|x| = x, if x ≥ 0
|x| = -x, if x < 0
Quick Examples
|7| = 7
7 is 7 units from 0
|-7| = 7
-7 is also 7 units from 0
|0| = 0
0 is 0 units from 0
3Properties of Absolute Value
These properties are crucial for simplifying expressions and solving equations and inequalities involving absolute value.
1. Non-negativity
|x| ≥ 0 for all real numbers x
Absolute value is always zero or positive.
2. Zero Property
|x| = 0 if and only if x = 0
Only zero has an absolute value of zero.
3. Symmetry
|x| = |-x|
A number and its opposite have the same absolute value. Example: |5| = |-5| = 5.
4. Multiplication Property
|ab| = |a| · |b|
Example: |3 × (-4)| = |-12| = 12. Also |3| × |-4| = 3 × 4 = 12.
5. Division Property
|a/b| = |a| / |b|, provided b ≠ 0
Example: |10 / (-2)| = |-5| = 5. Also |10| / |-2| = 10 / 2 = 5.
6. Square Root Property
√(x²) = |x|
The principal square root of x-squared equals the absolute value of x, not just x. Example: √((-3)²) = √9 = 3 = |-3|.
Drag the slider to see how absolute value measures the distance from zero, regardless of direction.
|0| = 0
Zero is 0 units from itself.
4Solving Absolute Value Equations
The key to solving absolute value equations is remembering that the expression inside the bars could be either positive or negative.
General Rule (where k ≥ 0)
If |expression| = k, then expression = k OR expression = -k
Steps
- Isolate the absolute value expression on one side.
- Set up two equations: one positive, one negative.
- Solve each equation independently.
- Check your solutions in the original equation.
Example: Solve |2x - 1| = 7
Absolute value is already isolated. Set up two cases:
Case 1 (Positive):
2x - 1 = 7
2x = 8
x = 4
Case 2 (Negative):
2x - 1 = -7
2x = -6
x = -3
Check: |2(4) - 1| = |7| = 7 ✓ and |2(-3) - 1| = |-7| = 7 ✓
Special Cases
|expression| = 0
Exactly one solution: expression = 0
|expression| = negative
No solution!
An absolute value can never equal a negative number.
5Solving Absolute Value Inequalities
Whether the absolute value is less than or greater than a number determines the type of inequality you set up. Visualizing on a number line helps immensely.
Less Than (< or ≤)
|expr| < k → -k < expr < k
Compound AND inequality. x is between -k and k.
Think: "Less than means between."
Greater Than (> or ≥)
|expr| > k → expr < -k OR expr > k
Compound OR inequality. x is outside the range.
Think: "Greater than means outside."
Example 1 (Less Than): Solve |3x + 2| ≤ 7
-7 ≤ 3x + 2 ≤ 7 (apply "less than" rule)
-9 ≤ 3x ≤ 5 (subtract 2)
-3 ≤ x ≤ 5/3
Solution: [-3, 5/3]
Example 2 (Greater Than): Solve |x - 4| > 3
Case 1:
x - 4 > 3
x > 7
Case 2:
x - 4 < -3
x < 1
Solution: (-∞, 1) ∪ (7, ∞)
Special Cases
|expr| < negative
No solution. Absolute value is never less than a negative number.
|expr| > negative
All real numbers. Absolute value is always ≥ 0, so always greater than any negative.
6Graphing Absolute Value Functions
The basic absolute value function y = |x| produces a V-shaped graph with its vertex at the origin (0, 0).
Vertex
(0, 0)
Axis of Symmetry
The y-axis (x = 0)
Slopes
+1 for x > 0, -1 for x < 0
Standard Form: y = a|x - h| + k
This form lets you identify all transformations from the basic y = |x| graph.
Vertex: (h, k)
h shifts horizontally (x - h shifts right h units). k shifts vertically (+k shifts up).
The "a" parameter
|a| > 1: vertically stretched (narrower V). 0 < |a| < 1: vertically compressed (wider V). a < 0: reflected across x-axis (opens downward).
Example: Graph y = |x + 1| - 2
h = -1 (because x - (-1) = x + 1)
k = -2
a = 1 (no reflection, no stretch)
Vertex: (-1, -2), opens upward
From the vertex, go up 1 and right 1 to (0, -1). Go up 1 and left 1 to (-2, -1). Draw the V-shape.
Example: Graph y = -½|x - 3| + 1
h = 3, k = 1
a = -1/2 (reflected, compressed)
Vertex: (3, 1), opens downward
The negative a means it opens downward. |a| = 1/2 means the V is wider. From the vertex, go down 1 and right 2 to (5, 0). Go down 1 and left 2 to (1, 0).
7Absolute Value & Distance
Absolute value is the mathematical tool for representing distance on a number line.
Distance from Zero
|x| = distance of x from 0
Distance Between Two Points
|a - b| = distance between a and b
Distance Inequalities
|x - c| < r
"x is within r units of c"
c - r < x < c + r
|x - c| > r
"x is more than r units from c"
x < c - r OR x > c + r
A factory produces bolts that should be 10 cm long, with a tolerance of 0.5 cm. Acceptable bolt lengths L satisfy |L - 10| ≤ 0.5, which means 9.5 ≤ L ≤ 10.5 cm.
8Worked Examples
Example 1: Multi-Step Equation
Solve: 3|x - 2| + 5 = 14
3|x - 2| = 9 (subtract 5)
|x - 2| = 3 (divide by 3)
x - 2 = 3 → x = 5
x - 2 = -3 → x = -1
Solutions: x = 5, x = -1
Example 2: No Solution
Solve: |4x + 8| = -12
No solution — absolute value cannot equal a negative number.
Example 3: AND Inequality
Solve: |2x + 5| - 1 < 6
|2x + 5| < 7 (add 1)
-7 < 2x + 5 < 7 (less than rule)
-12 < 2x < 2 (subtract 5)
-6 < x < 1
Solution: (-6, 1)
Example 4: OR Inequality
Solve: |x/3 - 2| ≥ 4
Case 1:
x/3 - 2 ≥ 4
x/3 ≥ 6
x ≥ 18
Case 2:
x/3 - 2 ≤ -4
x/3 ≤ -2
x ≤ -6
Solution: (-∞, -6] ∪ [18, ∞)
Example 5: Graphing
Graph: y = -½|x - 3| + 1
Vertex: (3, 1)
a = -1/2 → opens downward, wider V
Point right: (5, 0)
Point left: (1, 0)
V-shape opening downward through (1, 0), (3, 1), (5, 0)
9Common Mistakes
Forgetting the negative case
The most frequent error. When solving |expression| = k, you must consider both expression = k AND expression = -k. Missing the second case loses half your solutions.
Distributing into absolute value bars
|2x + 4| is NOT |2x| + |4|. You cannot distribute inside absolute value. However, you can factor: |2x + 4| = |2(x + 2)| = 2|x + 2| using the multiplication property.
Mixing up AND vs OR inequalities
|x| < k is AND (between): -k < x < k. |x| > k is OR (outside): x < -k or x > k. Remember: "Less thAND, greator."
Not isolating the absolute value first
Before applying any rules, get the | | term by itself. For 2|x| + 1 = 7, first subtract 1, then divide by 2 to get |x| = 3.
Assuming √(x²) = x
√(x²) = |x|, not x. For example, √((-3)²) = √9 = 3, not -3. The principal square root is always non-negative.
Quick Revision Summary
- ✓Absolute value measures a number's distance from zero and is always non-negative.
- ✓Piecewise definition: |x| = x if x ≥ 0, |x| = -x if x < 0.
- ✓Key properties: |x| = |-x|, |ab| = |a||b|, |a/b| = |a|/|b|, √(x²) = |x|.
- ✓Equations: If |expr| = k (k ≥ 0), then expr = k OR expr = -k.
- ✓Inequalities: less than = AND (between), greater than = OR (outside).
- ✓The graph of y = a|x - h| + k is a V-shape with vertex at (h, k).
- ✓If a > 0 the V opens up; if a < 0 it opens down. |a| controls width.
- ✓Distance between two points a and b: |a - b|.
- ✓Always isolate the absolute value before applying rules.
- ✓|expr| = negative has no solution; |expr| > negative is all reals.
Frequently Asked Questions
- Why is absolute value always positive?
- Because it represents distance. Distance is a scalar quantity that only tells you "how much" without direction. You cannot walk a negative distance. The absolute value of any number is its distance from zero on the number line, which is always zero or positive.
- Can an absolute value ever be zero?
- Yes. |x| = 0 only when x = 0. This means the number is 0 units away from 0, which is itself. Zero is the only number whose absolute value is zero.
- How is |x + 2| different from (x + 2)?
- |x + 2| evaluates to x + 2 when x + 2 is non-negative (x >= -2), and to -(x + 2) when x + 2 is negative (x < -2). Parentheses (x + 2) simply group the terms; their value is always x + 2 regardless of sign.
- What is the most common mistake students make with absolute value?
- Forgetting to consider both the positive and negative cases when solving equations or inequalities. When you have |expression| = k, you must set up expression = k AND expression = -k. Forgetting to isolate the absolute value term first is a close second.
- Where is absolute value used in real life?
- Absolute value appears in measurement and error analysis (e.g., a temperature tolerance of |T - 20| <= 2), distance calculations, computer science algorithms for calculating differences, and physics for the magnitude of vectors like speed (the absolute value of velocity).
Practice Quiz
Test your understanding of absolute value — select the correct answer for each question.
1.What is | -12 |?
2.Solve: |x| = 8
3.Solve: |x| < 3
4.Solve: |x| > 5
5.Solve: |x - 2| = 4
6.What is the vertex of y = |x + 3|?
7.Solve: |2x| = 6
8.Solve: |x + 1| < 2
9.What is |0|?
10.Solve: |x| > -1
Final Study Advice
- 1.Always isolate the absolute value before splitting into cases — it is the single most important step.
- 2.Remember the mnemonic: "Less thAND, greatOR" for inequality direction.
- 3.Sketch a number line when solving inequalities — it makes the solution set obvious.
- 4.For graphing, always find the vertex first, then use the slope from a to plot additional points.
- 5.Always check your solutions by substituting back into the original equation.