Lesson 1 — Properties of Inequality
Big idea: Inequalities compare unequal quantities. The same algebra rules apply as with equations, except multiplying or dividing by a negative reverses the inequality.
Words to know
| Word | Meaning |
|---|---|
| Inequality | A statement that one quantity is greater than or less than another: a > b, a < b. |
| Three possibilities property | For two quantities, exactly one is true: a > b, a = b, or a < b. |
| Transitive property | If a > b and b > c, then a > c. |
| Addition property | If a > b, then a + c > b + c. |
| Subtraction property | If a > b, then a − c > b − c. |
| Multiplication property | If a > b and c > 0, then ac > bc. |
| Division property | If a > b and c > 0, then a/c > b/c. |
| Whole greater than part | A whole is greater than each of its proper parts. |
Concepts to know
- Adding or subtracting the same number preserves the inequality.
- Multiplying or dividing by a positive number preserves the inequality.
- Multiplying or dividing by a negative number reverses the inequality.
- You can add inequalities that point the same way: if a > b and c > d, then a + c > b + d.
- If a > b and b > c, then a > c.
- The “whole greater than part” theorem is useful for angles and segments made of smaller parts.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Solve an inequality | Use the same steps as an equation; reverse the sign when multiplying/dividing by a negative. |
| Compare two lengths | Use given inequalities and the transitive property. |
| Justify a step | Name the property used: transitive, addition, subtraction, multiplication, division. |
| Combine inequalities | Add them only if the inequality signs point the same way. |
| Compare a whole and a part | Use “whole greater than part.” |
| Check an answer | Test a number that satisfies the inequality. |
Worked example
Solve 3x + 2 > 11.
3x > 9 Subtraction
x > 3 Division by positive 3If the inequality were −3x > 9, dividing by −3 would give x < −3.
Watch out
- Multiplying or dividing by a negative flips the inequality sign.
- Do not add inequalities that point in opposite directions.
- “Whole greater than part” applies to proper parts, not the whole itself.
- An inequality has many solutions, not just one.