An equation like $x = 3$ has one solution. An inequality like $x > 3$ has infinitely many: 3.1, 4, 7.5, 100, … You can order numbers, so you can picture all of them at once on a number line.
Key idea
$x > 3$ means every number greater than 3. To graph it, draw an open circle at 3 (3 is not included) and shade to the right. For $\ge$ or $\le$, the circle is closed (filled), because the boundary is included.
Why it works
There are too many solutions to list, so we draw them: the shaded ray holds every solution. The circle answers one small question: is the boundary itself a solution? For $x > 3$, try 3: $3 > 3$ is false, so the circle is left open. For $x \ge 3$, $3 \ge 3$ is true, so the circle is filled in.
| Symbol | Words | Circle | Shade |
|---|---|---|---|
| $x > 3$ | greater than 3 | open | right |
| $x \ge 3$ | at least 3 | closed | right |
| $x < 3$ | less than 3 | open | left |
| $x \le 3$ | at most 3, no more than 3 | closed | left |
Worked example: Graph it
Graph $x \le -1$.
- $\le$ includes $-1$, so draw a closed circle at $-1$.
- Less than means to the left: shade left from $-1$.
Answer: A closed circle at $-1$ with the ray shaded to the left.
Worked example: Words to an inequality
Riders must be at least 48 inches tall. Write an inequality for a rider’s height $h$.
- "At least 48" means 48 or more.
- So $h \ge 48$.
Answer: $h \ge 48$
Worked example: Read it from the variable
What does $3 < x$ mean?
- Read it right to left, starting with $x$: $x$ is greater than 3.
- So $3 < x$ is the same as $x > 3$. The pointed end of the sign always faces the smaller side.
Answer: $x$ is greater than 3.
Common mistake
Using a closed circle for $>$, or reading $3 < x$ as "$x$ is less than 3". Quick check: test a number. Is 3 a solution of $x > 3$? $3 > 3$ is false, so the circle must be open. Is 5 a solution of $3 < x$? $3 < 5$ is true, so the shading goes toward 5, to the right.