Introduction: What Are Inequalities?
This section introduces inequalities as mathematical sentences that describe relationships between two expressions that are not necessarily equal. Unlike equations (which use the = symbol), inequalities use the following symbols:
< means "less than"
> means "greater than"
≤ means "less than or equal to"
≥ means "greater than or equal to"
You'll learn how to:
Recognize inequality symbols.
Identify whether an expression is an inequality.
Translate verbal expressions (like “no more than”) into inequalities.
This chapter focuses on turning real-world scenarios into inequality statements. Examples include:
Understanding phrases like “at least,” “no more than,” and “less than.”
Modeling word problems using inequalities.
Interpreting the meaning behind inequality statements in practical contexts (e.g., prices, time limits, distances).
Here, students learn how to:
Visually represent inequalities on a number line.
Use open and closed circles to show whether endpoints are included:
Open circle: < or >
Closed circle: ≤ or ≥
Shade correctly to show all values that satisfy the inequality.
This chapter teaches how to solve inequalities involving:
Addition
Subtraction
Multiplication
Division
Key points:
The inequality sign must be preserved unless multiplying/dividing both sides by a negative number—in that case, the inequality direction flips.
The solution is often written with the inequality and shown on a number line.
Students now solve inequalities that require two operations:
Example: 2x+3<112x + 3 < 11
Process: First undo addition/subtraction, then multiplication/division.
Like in Chapter 3, signs flip when multiplying or dividing by negative numbers.
This chapter reinforces:
Careful use of inverse operations.
Step-by-step solving.
Graphing final answers for clarity.
The final chapter introduces compound inequalities, which combine two inequalities using:
AND (both conditions must be true):
e.g., 4<x≤94 < x \leq 9
OR (at least one condition must be true):
e.g., x<3x < 3 or x>7x > 7
Key learning:
How to solve and graph AND vs. OR inequalities.
Interpreting compound statements in real-life contexts.
Understanding inequality symbols and meanings.
Translating between words and math.
Solving and graphing both simple and complex inequalities.
Using inequalities to model everyday situations.