Linear Inequalities with One Variable - Comprehensive Guide

What is a Linear Inequality?
Inequality Symbols
Transformation Rules
Step-by-Step Solving
Writing Solutions
Special Cases
Comparison with Equations
Common Mistakes to Avoid
Practice Exercises
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1. What is a Linear Inequality?

A linear inequality with one variable is a mathematical statement that can be written in one of the following forms:

a x + b > c a x + b < c a x + b \geq c a x + b \leq c

where:

$a$ , $b$ , and $c$ are known numbers (coefficients)
$x$ is the unknown variable we want to find
The highest power of $x$ is 1 (that is what makes it "linear")

Difference from an Equation

With an equation like $2 x + 3 = 11$ , we look for one specific number ( $x = 4$ ).

With an inequality like $2 x + 3 > 11$ , we look for an entire set of numbers that satisfy the inequality ( $x > 4$ , meaning 5, 6, 4.1, 100, and so on).

Key difference: The solution to an equation is usually a single number. The solution to an inequality is an interval or set of numbers.

Examples of Linear Inequalities

Inequality	Type
$2 x + 3 > 11$	strict inequality (greater than)
$x - 5 < 8$	strict inequality (less than)
$3 x \geq 15$	non-strict inequality (greater than or equal to)
$\frac{x}{4} \leq 6$	non-strict inequality (less than or equal to)

What is NOT a Linear Inequality?

$x^{2} + 3 > 12$ --- $x$ has power 2 (quadratic inequality)
$2^{x} < 16$ --- $x$ is in the exponent (exponential inequality)
$\frac{1}{x} \geq 4$ --- $x$ is in the denominator (rational inequality)

2. Inequality Symbols

Symbol	Name	Meaning	Example
$<$	less than	left side is smaller than right side	$3 < 5$
$>$	greater than	left side is larger than right side	$7 > 2$
$\leq$	less than or equal to	left side is smaller or equals the right	$3 \leq 5$ , also $3 \leq 3$
$\geq$	greater than or equal to	left side is larger or equals the right	$7 \geq 2$ , also $7 \geq 7$

Strict vs. Non-Strict Inequalities

Strict inequalities ( $<$ , $>$ ) --- the boundary value is not included in the solution
Non-strict inequalities ( $\leq$ , $\geq$ ) --- the boundary value is included in the solution

Example: For $x > 3$ the number 3 is not a solution. For $x \geq 3$ the number 3 is a solution.

3. Transformation Rules

Rule 1: Addition and Subtraction

You can add or subtract any number on both sides of an inequality. The direction of the inequality does not change.

x + 5 > 12 \Rightarrow x + 5 - 5 > 12 - 5 \Rightarrow x > 7

x - 3 \leq 10 \Rightarrow x - 3 + 3 \leq 10 + 3 \Rightarrow x \leq 13

Rule 2: Multiplication and Division by a Positive Number

You can multiply or divide both sides of an inequality by a positive number. The direction of the inequality does not change.

3 x > 15 \Rightarrow \frac{3 x}{3} > \frac{15}{3} \Rightarrow x > 5

\frac{x}{4} \leq 6 \Rightarrow x \leq 24

Rule 3: Multiplication and Division by a Negative Number

WARNING! This is the most important rule! When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality!

- 2 x > 6 \Rightarrow \frac{- 2 x}{- 2} < \frac{6}{- 2} \Rightarrow x < - 3

- x \leq 5 \Rightarrow x \geq - 5

Why does the direction reverse? Consider concrete numbers:

We know that $3 < 7$
Multiply both sides by $- 1$ : we get $- 3$ and $- 7$
On the number line $- 3 > - 7$ , so the inequality flipped!

Rules Summary

Operation	Inequality Direction
$+ a$ or $- a$	stays the same
$\times a$ or $\div a$ where $a > 0$	stays the same
$\times a$ or $\div a$ where $a < 0$	reverses!

4. Step-by-Step Solving

Example 1: Simple Inequality

Solve: $2 x + 3 > 11$

Step 1: Subtract 3 from both sides

2 x + 3 - 3 > 11 - 3

2 x > 8

Step 2: Divide both sides by 2 (positive, direction stays the same)

\frac{2 x}{2} > \frac{8}{2}

x > 4

Step 3: Check by substitution (e.g.

x = 5

)

2 \cdot 5 + 3 = 13 > 11 ✓

Solution:

x \in (4, \infty)

Example 2: Inequality with a Negative Coefficient

Solve: $- 3 x + 6 \leq 15$

Step 1: Subtract 6 from both sides

- 3 x + 6 - 6 \leq 15 - 6

- 3 x \leq 9

Step 2: Divide both sides by

- 3

(negative, reverse the direction!)

\frac{- 3 x}{- 3} \geq \frac{9}{- 3}

x \geq - 3

Step 3: Check by substitution (e.g.

x = 0

)

- 3 \cdot 0 + 6 = 6 \leq 15 ✓

Also check the boundary point ( $x = - 3$ ):

- 3 \cdot (- 3) + 6 = 9 + 6 = 15 \leq 15 ✓

Solution:

x \in [- 3, \infty)

Example 3: Variable on Both Sides

Solve: $5 x - 4 > 2 x + 8$

Step 1: Move terms with

x

to the left side (subtract

2 x

)

5 x - 2 x - 4 > 8

3 x - 4 > 8

Step 2: Move numbers to the right side (add 4)

3 x > 12

Step 3: Divide by 3 (positive, direction stays the same)

x > 4

Step 4: Check by substitution (e.g.

x = 5

)

5 \cdot 5 - 4 = 21 > 2 \cdot 5 + 8 = 18 ✓

Solution:

x \in (4, \infty)

5. Writing Solutions

Solutions of an inequality can be written in three ways:

Way 1: Inequality Notation

Write the solution as an inequality:

x > 3 x \leq - 2 x \geq 0

Way 2: Interval Notation

Write the solution as an interval:

Inequality	Interval
$x > 3$	$(3, \infty)$
$x \geq 3$	$[3, \infty)$
$x < 3$	$(- \infty, 3)$
$x \leq 3$	$(- \infty, 3]$

Note: With $\infty$ and $- \infty$ always use a round bracket (open end), because infinity is not a specific number.

Way 3: Number Line Representation

Strict inequality ( $<$ , $>$ ) --- open circle on the boundary point
Non-strict inequality ( $\leq$ , $\geq$ ) --- filled circle on the boundary point
An arrow shows the direction where the solutions lie

For more detail see Linear Inequalities - Number Line.

6. Special Cases

Case 1: No Solution

When solving leads to a false statement.

Solve: $2 x + 3 > 2 x + 7$

Subtract $2 x$ from both sides:

3 > 7

This is false! The inequality has no solution.

Solution: $x \in \emptyset$ (empty set)

Case 2: All Real Numbers

When solving leads to a true statement.

Solve: $2 x + 3 < 2 x + 7$

Subtract $2 x$ from both sides:

3 < 7

This is always true! Every real number satisfies the inequality.

Solution: $x \in (- \infty, \infty) = R$

Case 3: Strict vs. Non-Strict at the Boundary

Solve: $3 x + 5 > 3 x + 5$

Subtract $3 x + 5$ :

0 > 0

This is false! Solution: $x \in \emptyset$

But if we had: $3 x + 5 \geq 3 x + 5$

Subtract $3 x + 5$ :

0 \geq 0

This is true! Solution: $x \in R$

For more detail see Linear Inequalities - Special Cases.

7. Comparison with Equations

Solving inequalities is very similar to solving equations. You use the same techniques:

Equations	Inequalities
Add/subtract on both sides	Add/subtract on both sides
Multiply/divide on both sides	Multiply/divide on both sides
The $=$ sign never changes	The direction changes when multiplying/dividing by a negative!
Solution: a single number ( $x = 4$ )	Solution: an interval ( $x > 4$ )
Check: substitute and verify equality	Check: substitute and verify inequality

Tip: If you can solve linear equations, you can solve inequalities too. Just remember one extra rule --- when multiplying or dividing by a negative number, flip the inequality sign.

8. Common Mistakes to Avoid

Mistake 1: Forgetting to Reverse the Inequality

- 2 x > 6

Wrong: $x > - 3$

Correct: $x < - 3$ (dividing by a negative number reverses the sign!)

Mistake 2: Incorrect Interval Notation

For $x > 3$ :

Wrong: $[3, \infty)$ --- the square bracket means 3 is included in the solution

Correct: $(3, \infty)$ --- round bracket, because 3 is not a solution

Mistake 3: Bracket at Infinity

Wrong: $[3, \infty]$

Correct: $Inequalities - addition and subtraction - Simple inequalities

Inequalities - Number Line - Represent solutions on a number line
Inequalities - Intervals - Write solutions as intervals
Inequalities - Both Sides - Variable on both sides
Inequalities - With Parentheses - Expanding brackets
Inequalities - With Fractions - Working with fractions
Inequalities - Special Cases - No solution or all real numbers
Inequalities - Mixed - Mixed practice

Linear Inequalities - Number Line - Graphical representation of solutions
Linear Inequalities - Intervals and Sets - Interval and set notation
Linear Inequalities - Rules and Formulas - Quick reference card
Linear Inequalities - Simple Inequalities - Basic types step by step
Linear Inequalities - Variable on Both Sides - More complex inequalities
Linear Inequalities - With Parentheses - Expanding brackets
Linear Inequalities - With Fractions - Working with fractions
Linear Inequalities - Special Cases - No solution or infinitely many solutions
Linear Equations - Comprehensive Guide - Comparison with equations

Linear Inequalities with One Variable - Comprehensive Guide

Table of Contents

1. What is a Linear Inequality?

Difference from an Equation

Examples of Linear Inequalities

What is NOT a Linear Inequality?

2. Inequality Symbols

Strict vs. Non-Strict Inequalities

3. Transformation Rules

Rule 1: Addition and Subtraction

Rule 2: Multiplication and Division by a Positive Number

Rule 3: Multiplication and Division by a Negative Number

Rules Summary

4. Step-by-Step Solving

Example 1: Simple Inequality

Example 2: Inequality with a Negative Coefficient

Example 3: Variable on Both Sides

5. Writing Solutions

Way 1: Inequality Notation

Way 2: Interval Notation

Way 3: Number Line Representation

6. Special Cases

Case 1: No Solution

Case 2: All Real Numbers

Case 3: Strict vs. Non-Strict at the Boundary

7. Comparison with Equations

8. Common Mistakes to Avoid

Mistake 1: Forgetting to Reverse the Inequality

Mistake 2: Incorrect Interval Notation

Mistake 3: Bracket at Infinity

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