Compound inequalities combine two or more inequality statements to describe a set of numbers. They are useful when a value must meet more than one condition or when it can fall in one of several allowed regions. Graphing them on a number line makes the solution set easier to see.
The two main connectors are AND and OR, and they create different kinds of graphs.
Key Facts
- AND means intersection: the solution must satisfy both inequalities.
- OR means union: the solution can satisfy either inequality or both.
- a < x < b means x > a AND x < b, so the graph is between a and b.
- x < a OR x > b graphs two separate rays going away from the middle.
- Use an open circle for < or > and a closed circle for ≤ or ≥.
- When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
Vocabulary
- Compound inequality
- A compound inequality is a statement that joins two or more inequalities using AND or OR.
- Intersection
- An intersection is the set of values that are shared by two solution sets.
- Union
- A union is the set of values that are in one solution set, the other solution set, or both.
- Interval notation
- Interval notation is a compact way to write a set of numbers using parentheses, brackets, and endpoints.
- Endpoint
- An endpoint is a boundary value on a number line where a solution interval begins or ends.
Common Mistakes to Avoid
- Treating AND like OR is wrong because AND keeps only the overlap of the two solution sets, not every value from both graphs.
- Using a closed circle for < or > is wrong because strict inequalities do not include the endpoint.
- Forgetting to reverse the inequality sign when dividing by a negative number is wrong because multiplying or dividing by a negative changes the order of the numbers.
- Writing interval notation with the wrong bracket is wrong because parentheses mean the endpoint is not included, while brackets mean it is included.
Practice Questions
- 1 Solve and graph the compound inequality 2 < x + 5 ≤ 9. Write the answer in interval notation.
- 2 Solve and graph the compound inequality 3x - 4 < -10 OR 2x + 1 ≥ 7. Write the answer in interval notation.
- 3 Explain why the solution to x > 1 AND x < 5 is a single interval, but the solution to x < 1 OR x > 5 is two separate rays.