How to graph inequalities on a number line
Quick answer
Draw a circle at the boundary number: open for < or > (the number isn’t included), closed for ≤ or ≥ (it is included). Then shade the line in the direction of the solutions: left for less than, right for greater than.
An inequality like x > 3 has endless answers: 4, 3.5, 3.001, a million. A number line lets you show all of them with one circle and one arrow.
Open or closed circle?
| Symbol | Meaning | Circle |
|---|---|---|
| < | less than | Open |
| > | greater than | Open |
| ≤ | less than or equal to | Closed (filled in) |
| ≥ | greater than or equal to | Closed (filled in) |
A trick to remember: the “or equal to” symbols have an extra line underneath, and the closed circle has extra ink inside.
Example 1: x > 3
Open circle at 3 (because 3 itself isn’t greater than 3), shaded to the right.
Example 2: x ≤ −1
Closed circle at −1 (because −1 is allowed), shaded to the left.
Example 3: a compound inequality
−2 < x ≤ 4 means x is bigger than −2 and at most 4. Draw both circles and shade only between them.
For an “or” inequality, such as x < 0 or x ≥ 5, you draw two separate arrows pointing away from each other.
Solve first, then graph
If the inequality has more than just x, solve it like an equation first. There’s one extra rule: when you multiply or divide by a negative number, flip the sign.
−2x > 6
Divide both sides by −2 and flip: x < −3
Graph that with an open circle at −3, shaded to the left. Check it by testing a number: x = −4 gives −2(−4) = 8, and 8 > 6 is true.
Need to tidy up the expression before solving? See how to simplify algebraic expressions.
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