๐ Understanding Compound Inequalities: 'And' vs. 'Or'
Compound inequalities combine two or more inequalities into one statement. The keywords 'and' and 'or' dramatically change how we solve and interpret them.
โ Definition of 'And' Inequalities
An 'and' inequality, also known as a conjunction, requires both inequalities to be true simultaneously. The solution set includes only the values that satisfy both inequalities. Think of it as finding the overlap or intersection of the two individual solution sets.
For example, consider the compound inequality: $x > 2$ and $x < 5$. This means $x$ must be greater than 2 and less than 5. The solution includes all numbers between 2 and 5 (not including 2 and 5 themselves).
โ Definition of 'Or' Inequalities
An 'or' inequality, also known as a disjunction, requires at least one of the inequalities to be true. The solution set includes all values that satisfy either inequality, or both. It is a union of the individual solution sets.
For example, consider the compound inequality: $x < -1$ or $x > 3$. This means $x$ must be less than -1 or greater than 3. The solution includes all numbers less than -1 and all numbers greater than 3.
๐ 'And' vs. 'Or' Comparison Table
| Feature | 'And' Inequality (Conjunction) | 'Or' Inequality (Disjunction) |
|---|
| Definition | Requires both inequalities to be true. | Requires at least one inequality to be true. |
| Keywords | and, but | or |
| Solution Set | Intersection of the individual solution sets. | Union of the individual solution sets. |
| Graph Representation | The solution is the overlapping region on the number line. | The solution includes two separate regions on the number line. |
| Example | $2 < x < 5$ (can be written as $x > 2$ and $x < 5$) | $x < -1$ or $x > 3$ |
๐ Key Takeaways
- ๐ 'And' = Intersection: 'And' inequalities represent the intersection of two solution sets, meaning the values must satisfy both conditions.
- ๐ค 'Or' = Union: 'Or' inequalities represent the union of two solution sets, meaning the values must satisfy at least one of the conditions.
- ๐ Graphing: The graph of an 'and' inequality shows the overlapping region, while the graph of an 'or' inequality shows two separate regions.
- ๐งฎ Solving: Solve each inequality separately and then determine the intersection or union of the solutions based on whether it's an 'and' or 'or' statement.