## How to Graph Inequalities on a Number Line

- Solve the inequality to get the variable by itself.
- Graph the boundary point of the inequality.
- If the inequality symbol is \(<\) or \(>\), graph the boundary point as an open circle.
- If the inequality symbol is \(\leq\) or \(\geq\), graph the boundary point as a closed circle.

- Determine which side of the number line should be shaded.
- If the variable is less than the boundary point, the number line will be shaded to the left.
- If the variable is greater than the boundary point, the number line will be shaded to the right.

## Examples

Graph the solutions: \[x>-3\]

*Is the variable isolated?*

\[\purple x>-3\]

Yes, the variable \(x\) is all by itself on one side of the inequality symbol. So, I don’t have to solve the inequality.

*Is the boundary point an open circle or a closed circle?*

\[\purple x>-3\]

The inequality symbol is \(>\), so I will graph the boundary point as an open circle.

*Which side of the number line should be shaded?*

\[\purple x>-3\]

The \(x\) is greater than -3, so I will shade the number line to the right.

The graph of \(\purple x>-3\) is…

Graph the solutions: \[5 \leq -b\]

*Is the variable isolated?*

No, the variable \(b\) has a negative sign with it.

\[\yellow 5 \leq -b\]

So, I need to solve the inequality by dividing both sides by -1. When I divide both sides of an inequality by a negative number, the inequality switches direction.

\[\yellow -5 \geq b\]

*Is the boundary point an open circle or a closed circle?*

\[\yellow -5 \geq b\]

The inequality symbol is \(\geq\), so I will graph the boundary point as a closed circle.

*Which side of the number line should be shaded?*

\[\yellow -5 \geq b\]

5 is greater than or equal to \(b\), which means that \(b\) is less than or equal to -5. So, I will shade the number line to the left.

The graph of \(\yellow 5 \leq -b\) is…

Graph the solutions: \[-2a+32<6a\]

*Is the variable isolated?*

No, the variable \(a\) is on two different sides of the inequality.

\[\red -2a+32<6a\]

So, I need to solve the inequality by adding 2a to both sides of the inequality.

\[\red 32<8a\]

Then I will divide both sides of the inequality by 8 to get \(a\) by itself.

\[\red 4<a\]

*Is the boundary point an open circle or a closed circle?*

\[\red 4<a\]

The inequality symbol is \(<\), so I will graph the boundary point as an open circle.

*Which side of the number line should be shaded?*

\[\red 4<a\]

4 is less than \(a\), which means that \(a\) is greater than 4. So, I will shade the number line to the right.

The graph of \(\red -2a+32<6a\) is…

Graph the solutions: \[4y+3 \geq 15\]

*Is the variable isolated?*

No, the variable \(y\) is multiplied by 4 and added to 3.

\[\blue 4y+3 \geq 15\]

So, I need to solve the inequality by subtracting 3 from both sides of the inequality.

\[\blue 4y \geq 12\]

Then I will divide both sides of the inequality by 4 to get \(y\) by itself.

\[\blue y \geq 3\]

*Is the boundary point an open circle or a closed circle?*

\[\blue y \geq 3\]

The inequality symbol is \(\geq\), so I will graph the boundary point as a closed circle.

*Which side of the number line should be shaded?*

\[\blue y \geq 3\]

The \(y\) is greater than or equal to 6. So, I will shade the number line to the right.

The graph of \(\blue 4y+3 \geq 15\) is…

## How to Write Inequalities from a Graph

- Identify the boundary point.
- Write down the variable and the value of the boundary point with a \(<\) or \(>\) in the middle.
- If the number line is shaded to the left, the inequality will look like this: \(x<4\)
- If the number line is shaded to the left, the inequality will look like this: \(x>4\)

- Determine whether the inequality symbol will have an “or equal to” line under it.
- If the boundary point is an open circle, keep the inequality symbol as \(<\) or \(>\).
- If the boundary point is an closed circle, change the inequality symbol to \(\leq\) or \(\geq\).

## Examples

Write an inequality for the solutions shown on this number line:

*What is the boundary point?*

1 is the boundary point between the shaded and unshaded sides of the number line.

*Is the number line shaded to the left or the right?*

The number line is shaded to the left so I can write my inequality as…

\[\yellow x < 1\]

*Does the inequality need an “or equal to” line under it?*

No, the boundary point is an open circle, so I will leave the inequality symbol as \(<\).

\(\yellow x<1\) is the most simplified inequality that describes the solutions shown on the number line.

Write an inequality for the solutions shown on this number line:

*What is the boundary point?*

-3 is the boundary point between the shaded and unshaded sides of the number line.

*Is the number line shaded to the left or the right?*

The number line is shaded to the right so I can write my inequality as…

\[\green x > -3\]

*Does the inequality need an “or equal to” line under it?*

Yes, the boundary point is a closed circle, so I will change the inequality symbol from\(>\) to \(/geq\).

\[\green x \geq -3\]

\(\green x \geq -3\) is the most simplified inequality that describes the solutions shown on the number line.