Graphing Inequalities on a Number Line

Q: When is the circle open and when is it closed?

Open for the strict symbols $ $, closed for $\leq$ and $\geq$. The circle is only about whether the boundary number itself counts as a solution.

Q: How do I remember which way to shade?

Test an easy number. For $x > 3$, try $5$: is $5 > 3$ true? Yes — and $5$ is to the right of $3$, so shade right. One test number settles the direction every time.

Q: What if the inequality is written like $5 < x$?

Read it toward the variable: $5 5$ — open circle at $5$, shade right.

An equation like $x = 3$ has one answer, so its graph is one dot. An inequality like $x > 3$ has infinitely many answers — every number bigger than $3$ — so its graph is a whole stretch of the number line.

Two choices tell the whole story: what kind of circle to draw at the boundary number, and which direction to shade. Get those two right and every one of these graphs is easy.

Open or closed circle

The circle sits at the boundary number and answers one question: is that number itself included? An open circle means not included — used with the strict symbols $<$ and $>$ . A closed, filled-in circle means included — used with $\leq$ and $\geq$ .

So $x > 3$ gets an open circle at $3$ , because $3$ itself is not a solution. $x \geq 1$ gets a closed circle at $1$ , because $1$ is a solution.

Which way to shade

Shade toward the numbers that make the inequality true. For $x > 3$ , the solutions are bigger than $3$ , and bigger numbers live to the right — so the ray points right. For $x \leq -2$ , shade left from $-2$ .

Here is $x \geq -2$ : a closed circle at $-2$ , with the shading running right toward the larger numbers.

When the variable is on the right

$-1 \geq x$ looks backwards, but it graphs like any other inequality once you read it toward the variable: it says $x$ is less than or equal to $-1$ . That's the same as $x \leq -1$ — closed circle at $-1$ , shade left.

A quick way to keep it straight: the symbol always opens toward the bigger side. If the open side faces away from the variable, the variable holds the smaller values.

Worked examples

Example 1: a strict inequality

Graph $x > 3$ on a number line.

The symbol

>

is strict, so the boundary is not included — open circle at

3

Solutions are greater than

3

, so shade to the right

Answer: Open circle at $3$ , ray pointing right

Example 2: less than or equal to

Graph $x \leq -2$ on a number line.

The symbol

\leq

includes the boundary — closed circle at

-2

Solutions are less than

-2

, so shade to the left

Answer: Closed circle at $-2$ , ray pointing left

Example 3: reading a graph

A number line shows a closed circle at $1$ with shading to the right. Which inequality does it represent?

Closed circle means the boundary number is included

\leq \text{ or } \geq

Shading to the right means values greater than

1

Combine the two observations

x \geq 1

Answer: $x \geq 1$

Try one yourself

Common questions

When is the circle open and when is it closed?

Open for the strict symbols $<$ and $>$ , closed for $\leq$ and $\geq$ . The circle is only about whether the boundary number itself counts as a solution.

How do I remember which way to shade?

Test an easy number. For $x > 3$ , try $5$ : is $5 > 3$ true? Yes — and $5$ is to the right of $3$ , so shade right. One test number settles the direction every time.

What if the inequality is written like

5 < x

Read it toward the variable: $5 < x$ says $x$ is greater than $5$ . Graph it as $x > 5$ — open circle at $5$ , shade right.

Why does the shading end in an arrow?

The arrow means the solutions keep going forever in that direction. There is no largest number greater than $3$ , so the graph of $x > 3$ can't stop — the arrow says so.

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