Building on Algebraic Expressions and Linear Equations

If you’ve already studied algebraic expressions and linear equations, you’re ready to explore an important and practical concept in algebra: inequalities.

In the previous article, you learned how to work with expressions like 3x + 5, and how to turn those into equations like 3x + 5 = 20. You already learned how to solve linear equations by isolating the variable. Now, we’ll go a step further and talk about algebraic inequalities—which look a lot like equations but instead of saying two sides are equal, they compare them.

What Is an Inequality?

An inequality is a mathematical sentence that compares two values or expressions using inequality symbols. It tells you that one side is either greater than, less than, or possibly equal to the other.

Common Inequality Symbols

• > means “greater than”
• < means “less than”
• ≥ means “greater than or equal to”
• ≤ means “less than or equal to”

Examples:

• x > 7 means x is greater than 7
• 3x + 1 < 10 means the expression 3x + 1 is less than 10
• x + 4 ≥ 12 means x + 4 is greater than or equal to 12

Unlike an equation (which usually has one specific solution), an inequality often has many possible solutions.

Solving Linear Inequalities

Solving an inequality is very similar to solving a linear equation, which you’ve already practiced. Your goal is still to isolate the variable (like x) on one side. You do this by using the same basic operations:

• Add or subtract the same value on both sides
• Multiply or divide both sides by the same number

Important Rule:
If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol.

Step-by-Step Examples

Example 1: Solve x + 5 < 12

Step 1: Subtract 5 from both sides
x < 7

This means any number less than 7 is a solution (e.g., 6, 0, -3).

Example 2: Solve 3x ≥ 15

Step 1: Divide both sides by 3
x ≥ 5

The solution includes 5 and any number greater than 5.

Example 3: Solve -2x > 10

Step 1: Divide both sides by -2
Important: Flip the inequality symbol!
x < -5

Graphing Inequalities on a Number Line

Inequalities can be represented visually on a number line, which helps you see the range of possible solutions.

Rules for Graphing:

• Use an open circle for < or > (the value itself is not included)
• Use a closed circle for ≤ or ≥ (the value is included)
• Shade the number line in the direction of the inequality

Example:
If x ≥ 2, you would:

• Put a closed circle on 2
• Shade to the right (toward bigger numbers)

Applications of Inequalities

Inequalities are used to describe real-world situations where more than one answer is possible.

Examples in Real Life:

• “You must be at least 16 years old to drive.” → x ≥ 16
• “You can spend no more than $50.” → x ≤ 50
• “A bag can hold less than 10 pounds.” → x < 10

Writing Inequalities from Word Problems

You can translate sentences into inequalities just like equations.

Example 1:

“A number minus 3 is more than 10.”
Let the number be x:
x - 3 > 10

Example 2:

“You must score at least 70 to pass.”
Let s be the score:
s ≥ 70

Practice Problems

1. x - 4 ≤ 6
2. 2x + 1 > 9
3. -x ≥ -8
4. “A number divided by 3 is less than 5.” → Write and solve the inequality.

Conclusion

Understanding inequalities helps you go beyond finding just one answer. Instead of showing where a variable is, inequalities show where it can be—often giving you a range of possible values.

Next, we’ll build on your knowledge of equations and graphs by exploring slope and rate of change—two important ideas that describe how variables are related. Understanding slope will help you better analyze linear relationships and the way things increase or decrease over time.