2.6 Solving Linear Inequalities

Learning Objectives

After completing this section, you will be able to:
  1. Express solutions in three equivalent forms:
    (1) Inequality notation   (2) Graph on a number line   (3) Interval notation
  2. Solve one-step, multi-step, and compound linear inequalities.
  3. Interpret the relationship between inequality symbols, brackets/parentheses, and shading direction.
  4. Recognize when to reverse the inequality sign.

Launch – Inequalities and Their Symbols

An inequality shows a range of possible values rather than one specific value.

Table 2.6
Symbol Meaning Example Description
< less than x < 3 x is less than 3
\le less than or equal to x \le 3 x is at most 3
> greater than x > 3 x is greater than 3
\ge greater than or equal to x \ge 3 x is at least 3

Key Rule

When multiplying or dividing by a negative number, reverse the inequality sign.

Graphing and Notation Styles

Each solution will be represented three ways:

  1. Inequality notation – the algebraic statement.
  2. Graph – a number line using parentheses ( ) or brackets [ ].
  3. Interval notation – symbolic form using the same parentheses/brackets.

Recall

Earlier algebra courses used open and closed circles on the number line.

In this course, we use:

  • Parentheses ( ) for open endpoints
  • Brackets [ ] for closed endpoints

Reading Graphs and Writing Inequalities

Each graph represents a solution set. Write:

  1. the inequality,
  2. the graph using parentheses or brackets, and
  3. the interval notation.

Example interpretations

Example 2.6.A

Inequality: x > 3

Graph:

Number line with an open parenthesis at 3 and shading to the right, representing the solution set x > 3.
Figure 2.6A

Interval: (3, \infty)

Strict inequalities (< or >) use parentheses. Infinity always uses a parenthesis.

Example 2.6.B

Inequality: x \ge -2

Graph:

Number line with a closed bracket at −2 and a shaded ray extending to the right, indicating all values greater than or equal to −2 (x ≥ −2).
Figure 2.6B

Interval: [-2, \infty)

Example 2.6.C

Inequality: x < -1

Graph:

Number line with an open parenthesis at −1 and shading extending left, representing x < −1
Figure 2.6C

Interval: (-\infty, -1)

Negative infinity is written as -\infty and always uses a parenthesis.

Example 2.6.D

Inequality: -2 < x \le 3 (compound inequality)

Graph:

Number line with an open parenthesis at −2 and a closed bracket at 3, shaded between −2 and 3, representing −2 < x ≤ 3.
Figure 2.6D

Interval: (-2, 3]

Try It

Solving Linear Inequalities

Solving a linear inequality is similar to solving a linear equation, but instead of finding one value, you find a range of values that makes the inequality true. Use inverse operations to isolate the variable, and remember to flip the inequality sign when you multiply or divide by a negative number.

Note: Inequalities are solved like equations, with one exception:

Whenever you multiply or divide by a negative number, you must flip the inequality sign.

Steps to Solve Linear Inequalities

  1. Simplify both sides.
  2. Add or subtract to isolate the variable term.
  3. Multiply or divide to solve for the variable.
  4. Reverse the inequality sign if you multiplied or divided by a negative.
  5. Express the solution in all three forms.

Example 2.6.1

Solve: x - 2 \le -4

Solution

Add 2 to both sides: x \le -2

Graph:

Number line with a closed bracket at −2 and shading extending left, representing x ≤ −2
Figure 2.6E

Interval: (-\infty, -2]

Example 2.6.2

Solve: -2x - 3 \ge 5

Solution

Add 3: -2x \ge 8

Divide by -2 (flip the sign): x \le -4

Graph:

Number line with a closed bracket at −4 and shading extending left, representing x ≤ −4
Figure 2.6F

Interval: (-\infty, -4]

Example 2.6.3

Solve: 2x - 3 \le 4x - 1

Solution

Subtract 2x from both sides: -3 \le 2x - 1

Add 1: -2 \le 2x

Divide by 2: -1 \le x, i.e., x \ge -1

Graph:

Number line with a closed bracket at −1 and a shaded ray extending to the right, indicating all values greater than or equal to −1 (x ≥ −1).
Figure 2.6G

Interval: [-1, \infty)

Example 2.6.4

Solve: -3(x - 2) < 2(x + 1)

Solution

Distribute: -3x + 6 < 2x + 2

Subtract 2 and add 3x: 4 < 5x

Divide by 5: 0.8 < x, i.e., x > 0.8

Graph:

Number line with an open parenthesis at 0.8 and shading to the right, representing x > 0.8
Figure 2.6H

Interval: (0.8, \infty)

Try It

Compound Inequalities

A compound (or double) inequality describes values between two numbers.

Example: -5 < x < 2 means x is greater than −5 AND less than 2.

Number line showing overlap between −5 and 2.
Figure 2.6I

Merging the overlap gives:

Number line shaded between −5 and 2 with open endpoints.
Figure 2.6J

Interval: (-5, 2)

General Form

When the inequality is written as a < x < b, the graph is shading between a and b with open endpoints:

Number line with open endpoints labeled a and b, shading between a and b.
Figure 2.6K

Example 2.6.5

Solve: -4 \le x - 3 \le 9

Solution

Add 3 to all three parts: -1 \le x \le 12

Graph:

Number line with shading from −1 to 12, closed at −1 and closed at 12.
Figure 2.6L

Interval: [-1, 12]

Example 2.6.6

Solve: -10 \le 5x < 20

Solution

Divide all three parts by 5: -2 \le x < 4

Graph:

Number line with shading from −2 to 4, closed at −2 and open at 4.
Figure 2.6M

Interval: [-2, 4)

Try It

Quick Reference

  • Parentheses ( ) → < or >
  • Brackets [ ] → \le or \ge

Try It – Summary

Key Takeaways

  1. Flip the inequality when multiplying or dividing by a negative.
  2. Parentheses mean strict inequality; brackets mean inclusive.
  3. Graphs, inequalities, and interval notation describe the same set.
  4. AND means “between.”
  5. Interval notation is the preferred format in College Algebra.

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Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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