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.
