Compound Inequality

What is a Compound Inequality?

A compound inequality is two inequalities combined by the words “and” or “or.” For example, $2 < x \leq 5$ means $x$ is greater than $2$ and less than or equal to $5$ .

Types of Compound Inequalities:

“And” Compound Inequality:
- Combines two conditions that must both be true at the same time.
- Represents the intersection of two inequalities, meaning the solution is where the conditions overlap.
- Example: $2 < x \leq 5$ means $x$ is greater than $2$ and less than or equal to $5$ . The solution is the interval $(2, 5]$ .
“Or” Compound Inequality:
- Combines two conditions where only one must be true.
- Represents the union of two inequalities, meaning the solution includes all values that satisfy either condition.
- Example: $x < -3$ or $x \geq 2$ means $x$ is less than $-3$ or greater than or equal to $2$ . The solution is the intervals $(-\infty, -3)$ and $[2, \infty)$ .

Graphical Representation:

“And” inequalities are shown as overlapping intervals on a number line, where the shaded region represents the intersection.
“Or” inequalities are shown as separate, non-overlapping shaded regions on a number line.

Examples in Real-World Contexts:

“And” Example:
- A teacher might set conditions for passing a test: “Students must score more than 50 but less than 80.” This is written as $50 < x < 80$ .
“Or” Example:
- A store might offer a discount: “Items are on sale if they cost less than $20 or more than$ 100.” This is written as $x < 20$ or $x > 100$ .

Solving Compound Inequalities:

Solve each part of the inequality separately.
Combine the solutions based on the “and” or “or” condition.
Graph the solution on a number line if needed.

Compound inequalities are a powerful tool for solving problems involving ranges and multiple conditions, making them essential in algebra and real-world applications.