Inequalities – When Values Have a Range
An inequality is like an equation, but instead of saying two things are equal, it says one is greater than, less than, or not equal to the other. Inequalities describe a range of solutions rather than a single value.
The Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| greater than | Strictly greater than | x greater than 3 means x could be 4, 5, 3.1 ... |
| less than | Strictly less than | x less than 7 means x could be 6, -1, 0 ... |
| greater than or equal to | Greater than or equal | x gte 5 includes x = 5 |
| less than or equal to | Less than or equal | x lte 2 includes x = 2 |
Solving Linear Inequalities
Solve inequalities the same way as equations — but remember: when you multiply or divide both sides by a NEGATIVE number, you must reverse the inequality sign.
Add 4: 3x is greater than 12. Divide by 3: x is greater than 4. Solution set: all numbers greater than 4.
Subtract 2: -5x is less than or equal to 10. Divide by -5 and REVERSE the sign: x is greater than or equal to -2.
Showing Solutions on a Number Line
| Inequality | Endpoint | Arrow Direction |
|---|---|---|
| x greater than 3 | Open circle at 3 (3 not included) | Arrow pointing right |
| x less than or equal to 5 | Closed circle at 5 (5 included) | Arrow pointing left |
Double Inequalities
A double inequality restricts x from both sides.
Subtract 3 throughout: -4 is less than 2x which is less than or equal to 4. Divide by 2: -2 is less than x which is less than or equal to 2. x can be any number between -2 (not included) and 2 (included).
Integer Solutions
Integers: -2, -1, 0, 1, 2. (Note: -3 is excluded because the sign is strictly less than.)
Real-Life Application
A lift can carry at most 500 kg. If the average person weighs 70 kg, the number of people n must satisfy 70n is less than or equal to 500, so n is less than or equal to 7.14 — meaning at most 7 people.
Common Mistakes
- Forgetting to reverse the inequality when multiplying or dividing by a negative number.
- Using a closed circle when the symbol is strictly less/greater than (should be open).
- Listing the wrong integers because the boundary is excluded.
Key Takeaways
- Inequalities describe a range of values, not a single answer.
- Solve like an equation, but reverse the sign when multiplying or dividing by a negative.
- Show solutions on a number line: open circles for strict inequalities, closed for inclusive ones.
Practice Questions
- Solve 4x + 2 is greater than 14.
- Solve 10 - 3x is less than 1.
- Solve and show on a number line: -2 is less than or equal to 3x + 1 which is less than 10.
- List all integer values of n where -4 is less than 2n is less than or equal to 6.
- A budget for food is at most 50 per week. Meals cost 8 each. Write and solve an inequality to find the maximum number of meals.