An inequality shows that two values are not equal — one is larger or smaller than the other. Instead of an equals sign, we use special symbols to show the relationship. Inequalities can also be shown on a number line, which gives a clear visual picture of all the values included.
1. Inequality Symbols #
| Symbol | Meaning | Example | Read as |
|---|---|---|---|
| $<$ | less than | $x < 5$ | $x$ is less than 5 |
| $>$ | greater than | $x > 3$ | $x$ is greater than 3 |
| $\leq$ | less than or equal to | $x \leq 4$ | $x$ is less than or equal to 4 |
| $\geq$ | greater than or equal to | $x \geq -2$ | $x$ is greater than or equal to −2 |
2. Representing Inequalities on a Number Line #
On a number line, a circle marks the boundary value and a line shows which values are included. The type of circle tells you whether the boundary is included or not.
| Circle | Meaning | Used for |
|---|---|---|
| Open circle | Boundary value is not included | $<$ and $>$ |
| Closed circle | Boundary value is included | $\leq$ and $\geq$ |
After placing the circle, draw the line in the correct direction:
- $x > a$ or $x \geq a$ → line goes to the right (larger values)
- $x < a$ or $x \leq a$ → line goes to the left (smaller values)
Greater than 2, but 2 is not included. Open circle at 2, line to the right.
Less than or equal to −1, and −1 is included. Closed circle at −1, line to the left.
3. Double Inequalities #
A double inequality has two boundary values and describes a range. For example, $-3 \leq x < 1$ means $x$ is at least −3 and less than 1.
- Left boundary: $-3 \leq$ → $\leq$ means inclusive → closed circle at −3
- Right boundary: $x < 1$ → $<$ means strict → open circle at 1
- Draw a line between the two circles
4. Interpreting a Number Line #
You can also work the other way — look at a number line and write the inequality it shows. Follow these steps:
- Find the boundary value (where the circle is)
- Check the circle type: open circle → strict ($<$ or $>$) | closed circle → inclusive ($\leq$ or $\geq$)
- Check the direction of the line: right → greater than | left → less than | between two circles → double inequality
- Open circle at 4 → 4 is not included → strict
- Line goes right → greater than
- Answer: $x > 4$
- Closed circle at −2 → −2 is included → $\leq$
- Open circle at 3 → 3 is not included → $<$
- Answer: $-2 \leq x < 3$
