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Absolute Value

Absolute value measures distance from zero , it always returns a non-negative result regardless of whether the input is positive or negative. It's one of the most frequently tested algebra concepts and appears constantly in higher math.

In this lesson

  1. What absolute value means
  2. The notation and basic calculations
  3. Solving absolute value equations
  4. Absolute value inequalities
  5. Real-world applications

1 What Absolute Value Means

The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, so absolute value is always ≥ 0.

|7| = 7 (7 is 7 units from zero). |−7| = 7 (−7 is also 7 units from zero). |0| = 0.

Think distance, not direction

Absolute value removes the sign , it only cares about magnitude. How far? Not which way. This is why |−7| = |7| = 7. Both are 7 steps from zero, just in opposite directions.

2 The Notation and Basic Calculations

Written with vertical bars: |x|. Formally defined as: |x| = x if x ≥ 0, and |x| = −x if x < 0.

That second part (|x| = −x when x is negative) confuses students. Example: |−5| = −(−5) = 5. You're negating a negative, which produces a positive. The absolute value is always positive.

Evaluating Absolute Values
Evaluate: |3 − 8| and |8 − 3|
1|3 − 8| = |−5| = 5
2|8 − 3| = |5| = 5
3Both give the same result , absolute value measures the distance between 3 and 8 regardless of direction
Answer: |3 − 8| = |8 − 3| = 5

3 Solving Absolute Value Equations

|x| = 5 has two solutions: x = 5 and x = −5. Both are 5 units from zero. When solving |expression| = c (where c > 0), split into two equations: expression = c and expression = −c.

Absolute Value Equation
Solve: |2x − 3| = 7
1Split into two cases:
2Case 1: 2x − 3 = 7 → 2x = 10 → x = 5
3Case 2: 2x − 3 = −7 → 2x = −4 → x = −2
4Check x=5: |2(5)−3| = |7| = 7 ✓
5Check x=−2: |2(−2)−3| = |−7| = 7 ✓
Answer: x = 5 or x = −2
|expression| = negative has no solution

If |x| = −3, there is no solution. Absolute value is always ≥ 0 , it can never equal a negative number.

4 Absolute Value Inequalities

|x| < c means x is within c units of zero: −c < x < c (AND compound inequality)

|x| > c means x is more than c units from zero: x < −c OR x > c (OR compound inequality)

Absolute Value Inequality
Solve: |x − 4| ≤ 3
1This means x − 4 is within 3 units of zero
2−3 ≤ x − 4 ≤ 3
3Add 4 to all parts: 1 ≤ x ≤ 7
Answer: x is between 1 and 7 inclusive , all values within 3 of 4

5 Real-World Applications

Manufacturing tolerances: a bolt specified at 10mm ± 0.5mm accepts any bolt where |diameter − 10| ≤ 0.5. This is an absolute value inequality describing the acceptable range.

Statistics: the absolute deviation of a value from the mean is |x − μ|. Mean absolute deviation (MAD) is the average of all absolute deviations , a simpler alternative to standard deviation.

Computer science: absolute value appears in distance calculations, error metrics (mean absolute error in machine learning), and comparison algorithms.

Practice Problems

Evaluate: |−12| + |5 − 9|
12 + |−4| = 12 + 4 = 16
Solve: |x + 3| = 10
Case 1: x + 3 = 10 → x = 7. Case 2: x + 3 = −10 → x = −13. Answer: x = 7 or x = −13
Solve: |2x| < 8
−8 < 2x < 8. Divide by 2: −4 < x < 4
Does |a + b| = |a| + |b| always hold?
No , only when a and b have the same sign. If a = 3 and b = −3: |3 + (−3)| = |0| = 0, but |3| + |−3| = 6. This is the triangle inequality: |a+b| ≤ |a|+|b|.

Sources & Further Reading

The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.

Khan AcademyAbsolute ValueVideo lessonsMath is FunAbsolute ValueClear explanationDesmosNumber LineVisualize absolute valueWikipediaAbsolute ValueFormal definition