Inequalities compare quantities that may be less than, greater than, or equal to each other. They are used whenever a problem has a range of possible answers instead of one exact value. Solving inequalities helps describe limits such as budgets, speeds, scores, and safe operating ranges.
A number line makes these solution sets visible and easy to check.
Key Facts
- Inequality symbols: < means less than, > means greater than, <= means less than or equal to, and >= means greater than or equal to.
- Solve linear inequalities using inverse operations, just like equations, while keeping the variable isolated.
- If you add or subtract the same number on both sides, the inequality sign does not change.
- If you multiply or divide both sides by a positive number, the inequality sign does not change.
- If you multiply or divide both sides by a negative number, flip the inequality sign, such as -2x < 8 becomes x > -4.
- Use an open circle for < or >, a closed circle for <= or >=, and shade toward all numbers that make the inequality true.
Vocabulary
- Inequality
- A mathematical statement that compares two expressions using symbols such as <, >, <=, or >=.
- Solution set
- The set of all values that make an inequality true.
- Number line
- A straight line used to show the order and location of real numbers.
- Open circle
- A circle on a number line showing that the endpoint is not included in the solution.
- Closed circle
- A filled circle on a number line showing that the endpoint is included in the solution.
Common Mistakes to Avoid
- Forgetting to flip the inequality sign when multiplying or dividing by a negative number. This is wrong because the order of the two sides reverses when both sides are scaled by a negative value.
- Using a closed circle for x < 3 or x > 3. This is wrong because strict inequalities do not include the endpoint.
- Shading the number line in the wrong direction. Test a simple value, such as 0, to check whether the shaded side actually satisfies the inequality.
- Changing the inequality sign when adding or subtracting. Addition and subtraction by the same amount do not reverse the order of the two sides.
Practice Questions
- 1 Solve and graph the solution on a number line: 3x + 5 <= 17.
- 2 Solve and graph the solution on a number line: -4x + 7 > 19.
- 3 A student solves -2x <= 10 and writes x <= -5. Explain the error and describe the correct graph.