A compound inequality is simply two or more inequalities combined together using the words "or" or "and". This combination creates a more complex condition that a variable must satisfy. The type of connector ("or" or "and") significantly impacts the solution set.
Understanding the Types of Compound Inequalities
Let's break down the two main types:
"Or" Inequalities
- Definition: An "or" inequality states that a solution must satisfy at least one of the inequalities. In other words, the value just needs to work for one part of the inequality to be considered a solution.
- Example:
x ≤ -3 or x > 5. Any number less than or equal to -3, or any number greater than 5, is a solution to this compound inequality. - Solution Set: The solution set will include all values that satisfy the first inequality combined with all values that satisfy the second inequality. This often results in two separate ranges on a number line.
"And" Inequalities
- Definition: An "and" inequality states that a solution must satisfy both inequalities simultaneously. The value must make all parts of the inequality true.
- Example:
-2 < x and x ≤ 4can be written as-2 < x ≤ 4. Here, 'x' must be greater than -2 and less than or equal to 4. - Solution Set: The solution set only includes the values that are present in the solution of both inequalities. The solution usually lies in the overlap (intersection) of the individual inequalities.
Key Differences Summarized
Here's a table outlining the crucial distinctions:
| Feature | "Or" Inequality | "And" Inequality |
|---|---|---|
| Connector | or | and |
| Solution Requirement | Must satisfy at least one inequality | Must satisfy both inequalities |
| Solution Set | Union of individual solution sets | Intersection (overlap) of individual solution sets |
Practical Example: Temperature Ranges
Imagine you're planning a picnic. You want the temperature to be either below 60°F or above 80°F. This could be represented as a compound inequality:
T < 60 or T > 80
If the forecast shows 55°F (T=55), your picnic is on! If it's 85°F (T=85), picnic is on as well! If it's 70°F (T=70), your picnic should be postponed because it doesn't fit either inequality.
Contrast this with finding the perfect temperature for a specific plant that must remain between 65°F and 75°F:
65 ≤ T and T ≤ 75, which could be written as 65 ≤ T ≤ 75
Here, the temperature must fit within both the lower and upper limits. 70°F would be fine, but 60°F or 80°F would not.
