Inequalities with Absolute Value

TipLearning Objectives

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

  • Solve absolute value inequalities of the form \(|x| < a\), \(|x| > a\), and their shifted/scaled versions.
  • Interpret solutions using number-line reasoning.
  • Express solutions using interval notation or compound inequalities.
  • Solve absolute value inequalities in real-world contexts.

Key Ideas

Absolute value measures distance from zero on the number line.

General forms:

  • Less than:
    \[ |x| < a \quad \Longrightarrow \quad -a < x < a \]

  • Less than or equal:
    \[ |x| \le a \quad \Longrightarrow \quad -a \le x \le a \]

  • Greater than:
    \[ |x| > a \quad \Longrightarrow \quad x < -a \ \text{ or }\ x > a \]

  • Greater than or equal:
    \[ |x| \ge a \quad \Longrightarrow \quad x \le -a \ \text{ or }\ x \ge a \]

For expressions like \(|x - h|\):

  • \(|x - h|\) measures distance from \(h\).
  • Inequalities describe intervals around or beyond \(h\).

Common Problem Types

Basic Less-Than Inequalities

Translate \(|x|<a\) into a double inequality.

Example:
\(|x| < 3 \Rightarrow -3 < x < 3\).

Basic Greater-Than Inequalities

“Greater than” becomes an OR compound inequality.

Example:
\(|x| > 5 \Rightarrow x < -5\) or \(x > 5\).

Shifted Absolute Value

\(|x - h|\) means distance from \(h\).

Example:
\(|x - 4| < 2 \Rightarrow 2 < x < 6\).

Scaled Absolute Value

First isolate the absolute value.

Example:
\(3|x + 1| \le 9 \Rightarrow |x + 1| \le 3\).

Real-World Context

Absolute value describes a tolerance or maximum deviation.

Example:
A machine allows a temperature within \(2^\circ\) of \(70^\circ\):
\[ |T - 70| \le 2 \]

Strategies

  • Always isolate the absolute value before solving.
  • Use distance interpretation: “within” → less than; “outside” → greater than.
  • Write compound inequalities clearly.
  • Check endpoints for strict vs. non-strict inequalities.
  • If needed, sketch a quick number line.

Worked Examples

Example 1 — Less Than

Solve: \[ |x| < 4 \]

Translate: \[ -4 < x < 4 \]

Example 2 — Greater Than with Shift

Solve: \[ |x - 3| > 5 \]

Interpret as distance > 5 from 3: \[ x < -2 \quad \text{or} \quad x > 8 \]

Example 3 — Isolate and Solve

Solve: \[ 2|x + 4| \le 10 \]

Divide both sides: \[ |x + 4| \le 5 \]

Translate: \[ -5 \le x + 4 \le 5 \]

Subtract 4: \[ -9 \le x \le 1 \]

Example 4 — Real-World Tolerance

A part must be within \(0.1\) units of length \(5\).

Model: \[ |L - 5| \le 0.1 \]

Thus: \[ 4.9 \le L \le 5.1 \]

WarningCommon Mistakes
  • Forgetting to isolate the absolute value before solving.
  • Mixing up when to use AND vs. OR.
  • Reversing inequalities incorrectly when moving terms.
  • Dropping negative signs when translating to compound inequality form.
  • Confusing strict (\(<\)) and non-strict (\(\le\)) boundaries.

Practice Problems

  1. Solve:
    \[ |x| \le 6 \]

  2. Solve:
    \[ |x - 2| > 3 \]

  3. Solve:
    \[ 4|x + 5| < 12 \]

  4. A measurement \(M\) must be within \(3\) units of \(20\). Write the inequality and solution interval.

1.
\[ -6 \le x \le 6 \]

2.
\[ x < -1 \quad \text{or} \quad x > 5 \]

3.
Divide: \(|x + 5| < 3\)
Translate: \[ -3 < x + 5 < 3 \] Subtract 5: \[ -8 < x < -2 \]

4.
Model: \(|M - 20| \le 3\)
Interval: \[ 17 \le M \le 23 \]

Summary

  • “Less than” absolute value → compound AND inequality.
  • “Greater than” absolute value → compound OR inequality.
  • Always isolate \(| \cdot |\) first.
  • Use number-line interpretation to double-check solutions.
  • Within = \(|x - h| < a\).
  • At least/at most = check inequality sign carefully.
  • For greater-than inequalities, split into two cases.
  • Draw a number line to visualize intervals.