Absolute Value Basics

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Understand absolute value as distance from zero on the number line.
  • Evaluate absolute value expressions correctly.
  • Interpret absolute value in equations and basic distance problems.
  • Work carefully with negative signs inside and outside absolute value bars.

Key Ideas

  • The absolute value of a number represents its distance from 0, so it is always non-negative.

\[ |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases} \]

Examples:

  • \(|5| = 5\)

  • \(|-5| = 5\)

  • \(|0| = 0\)

  • Absolute value is about distance, not “removing the minus sign.”

  • It applies to the entire expression inside the bars.

Number-line visualization showing |x| as the distance from 0.
Important

Absolute value bars act as grouping symbols.
Always simplify the expression inside the bars before applying absolute value.

Common Problem Types

1. Evaluating Expressions

Example:
\[ |3 - 7| \]

Step-by-step:

  1. Inside first → \(3 - 7 = -4\)
  2. Absolute value → \(|-4| = 4\)

2. Absolute Value With Multiplication

Example:
\[ |-2 \cdot 6| = |-12| = 12 \]

3. Negative Signs Outside Bars

Example:
\[ -|\, -8 \,| \]

Step-by-step:

  1. Inside: \(|-8| = 8\)
  2. Apply outer negative: \(-8\)

These are two separate operations.

4. Simple Absolute Value Equations

Example: Solve \(|x| = 7\).

Two cases: - \(x = 7\)
- \(x = -7\)

5. Distance on a Number Line

Distance between \(-3\) and \(5\):

\[ |5 - (-3)| = |8| = 8 \]

Number line showing distance between -3 and 5.

Strategies

  • Always evaluate inside the bars first.
  • Keep outside negatives separate.
  • Think of absolute value as distance, not a symbol that “makes numbers positive.”
  • Equations of the form \(|x| = a\) (with \(a>0\)) always yield two solutions.

Worked Examples

Example 1

Question:
\[ |2 - 9| \]

Solution:

  1. Inside: \(2 - 9 = -7\)
  2. Apply absolute value: \(|-7| = 7\)

Answer: \(7\)

Example 2

Question:
\[ -|4 - 10| \]

Solution:

  1. Inside: \(4 - 10 = -6\)
  2. Absolute value: \(|-6| = 6\)
  3. Apply outer negative: \(-6\)

Answer: \(-6\)

Example 3

Question: Solve
\[ |x - 3| = 5 \]

Solution:
Split into two cases:

  1. \(x - 3 = 5\) → \(x = 8\)
  2. \(x - 3 = -5\) → \(x = -2\)

Answer: \(x = 8\) or \(x = -2\)

WarningCommon Mistakes
  • Ignoring an outside negative sign.
  • Thinking \(|a+b| = |a| + |b|\) (not true).
  • Solving only one branch of an absolute value equation.
  • Forgetting to simplify inside the bars first.
  • Mixing absolute value with distribution incorrectly.

Practice Problems

  1. \(|7 - 12|\)
  2. \(-|\, -9 \,|\)
  3. \(|3x|\) when \(x = -4\)
  4. Solve \(|x| = 11\)
  5. Solve \(|x - 2| = 4\)

1. \(|7 - 12|\)
Inside → \(7 - 12 = -5\)
Absolute value → \(5\)
Answer: \(5\)

2. \(-|\, -9 \,|\)
Inside → \(|-9| = 9\)
Outside → \(-9\)
Answer: \(-9\)

3. \(|3x|\) when \(x = -4\)
Substitute: \(|3(-4)| = |-12| = 12\)
Answer: \(12\)

4. Solve \(|x| = 11\)
Two cases → \(x = 11\) or \(x = -11\)
Answer: \(x = \pm 11\)

5. Solve \(|x - 2| = 4\)
Case 1: \(x - 2 = 4\) → \(x = 6\)
Case 2: \(x - 2 = -4\) → \(x = -2\)
Answer: \(x = 6\) or \(x = -2\)

Summary

  • Absolute value measures distance from 0.
  • Bars act like grouping symbols—simplify inside first.
  • Watch for negative signs outside the bars.
  • Equations of the form \(|x - a| = b\) split into two linear equations.
  • Evaluate inside the bars first, then apply any outside operations.
  • For \(|x - a| = b\) with \(b>0\), expect two solutions.
  • Draw a quick number line to visualize distances.