Concept Overview
An inequality compares two expressions using symbols like \( < \), \(> \), \( \leq \), or \( \geq \). Solving an inequality means finding the values that make it true, and these values are often shown as a shaded region on a number line.
Just like equations, inequalities can be solved using inverse operations. But when multiplying or dividing both sides by a negative number, the inequality symbol must be reversed.
Why does the symbol flip? Think of inequalities as preserving a sense of “greater than” or “less than.” When you multiply or divide both sides by a negative number, the direction of that comparison is reversed. For example, \( 5 > 2 \) is true—but if we multiply both sides by \(-1\), we get \( -5 > -2 \), which is false. To keep the inequality true, we must flip the symbol: \( -5 < -2 \). That's why the rule exists—it ensures the relationship still makes sense.
Key Vocabulary
- Inequality: A mathematical sentence comparing expressions using \( <,>, \leq, \geq, \neq \)
- Solution Set: All values that make the inequality true
- Open Circle: Used to show values that are not included (e.g., \( < \) or \(> \))
- Closed Circle: Used to show values that are included (e.g., \( \leq \) or \( \geq \))
Review of Inequality Symbols
Inequality symbols are used to compare two values or expressions. They help us describe whether one quantity is smaller, larger, or possibly equal to another. Here’s a quick review:
| Symbol | Meaning | Example | Word Form |
|---|---|---|---|
| \( < \) | Less than | \( 3 < 7 \) | 3 is less than 7 |
| \( > \) | Greater than | \( 9 > 4 \) | 9 is greater than 4 |
| \( \leq \) | Less than or equal to | \( x \leq 5 \) | x is less than or equal to 5 |
| \( \geq \) | Greater than or equal to | \( y \geq -2 \) | y is greater than or equal to -2 |
| \( \neq \) | Not equal to | \( x \neq 3 \) | x is not equal to 3 |
Tip: Think of the inequality symbol as a hungry mouth—the open side always faces the larger number!
Solving Inequalities Step-by-Step
Solving an inequality is like solving an equation—you want to isolate the variable using inverse operations. But there’s one key difference: if you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
Problem: Solve \( -2x + 5 > 1 \)
- Subtract 5 from both sides: \( -2x > -4 \)
- Divide by -2 (and flip the sign): \( x < 2 \)
Quick Tips
- Use opposite operations to undo addition, subtraction, multiplication, or division.
- Keep the inequality balanced by doing the same thing to both sides.
- Reverse the inequality symbol only when multiplying or dividing by a negative number.
Interpreting Number Line Inequalities
Number lines are a powerful way to visualize inequality solutions. Each graph shows where the variable is allowed to be.
Greater than:
The solution begins just after -1 (open circle) and continues to the right.
Less than:
The solution ends just before 2 (open circle) and extends to the left.
Greater than or equal to:
\( x \geq -2 \)
The solution includes -2 (closed circle) and all values greater than it (arrow right).
Less than or equal to:
The solution includes 3 (closed circle) and all values less than it (arrow left).
Reminder:
- An open circle means the value is not included (\( < \) or \(> \))
- A closed circle means the value is included (\( \leq \) or \( \geq \))
Try It Yourself!
- Solve and graph: \( x – 2 \leq 6 \)
- Solve and graph: \( -4x \leq 8 \)
- Solve and graph: \( 5x + 3 > 18 \)
- Solve and graph: \( \frac{x}{2} \geq -1 \)
Reveal Answers
-
\( x \leq 8 \)
-
\( x \geq -2 \)
-
\( x > 3 \)
-
\( x \geq -2 \)
Connections & Extensions
Inequalities are used to describe ranges in budgets, measurements, and constraints in real-world systems. They are foundational for working with systems of inequalities and constraints in linear programming, data science, and engineering.