Integers and real numbers can be represented on a number line. The point on this line associated with each number is called the graph of the number. Notice that number lines are spaced equally, or proportionately (see Figure 1).
Figure 1. Number lines.
Graphing inequalities
When graphing inequalities involving only integers, dots are used.
Example 1
Graph the set of x such that 1 ≤ x ≤ 4 and x is an integer (see Figure 2).
{ x:1 ≤ x ≤ 4, x is an integer}
Figure 2. A graph of {x:1 ≤ x ≤ 4, x is an integer}.
When graphing inequalities involving real numbers, lines, rays, and dots are used. A dot is used if the number is included. A hollow dot is used if the number is not included.
Example 2
Graph as indicated (see Figure 3).
- Graph the set of x such that x ≥ 1.
- { x: x ≥ 1}
- Graph the set of x such that x > 1 (see Figure 4).
- { x: x > 1}
- Graph the set of x such that x < 4 (see Figure 5).
- { x: x < 4}
This ray is often called an open ray or a half line. The hollow dot distinguishes an open ray from a ray.
Figure 3. A graph of { x: x ≥ 1}.
Figure 4. A graph of { x: x > 1}
Figure 5. A graph of { x: x < 4}
Intervals
An interval consists of all the numbers that lie within two certain boundaries. If the two boundaries, or fixed numbers, are included, then the interval is called a closed interval. If the fixed numbers are not included, then the interval is called an open interval.
Example 3
Graph.
- Closed interval (see Figure 6).
- { x: –1 ≤ x ≤ 2}
- Open interval (see Figure 7).
- { x: –2 < x < 2}
Figure 6. A graph showing closed interval { x: –1 ≤ x ≤ 2}.
Figure 7. A graph showing open interval { x: –2 < x < 2}.
If the interval includes only one of the boundaries, then it is called a half‐open interval.
Example 4
Graph the half‐open interval (see Figure 8).
{ x: –1 < x ≤ 2}
Figure 8. A graph showing half‐open interval { x: –1 < x ≤ 2}.
Graphing Inequalities on a Number Line
Inequalities are exactly what they sound like: equations where the sides are "inequal" (not equal) to each other. There are five basic inequalities that we need to be familiar with:
| Symbol | Meaning |
| < | less than |
| > | greater than |
| ≤ | less than or equal to |
| ≥ | greater than or equal to |
| ≠ | not equal to |
The inequality y < 2 means that y can be any number less than 2 (such as 1.9, 0.75, 0, -6, etc.).
The inequality y > 7 means that y can be any number greater than 7 (such as 7.1, 8, 9, 537, etc.).
The inequality y ≤ 2 means that y can be any number less than 2, or it can be equal to 2 itself (2, 1.9, 1, 0, -6, etc.).
Last but not least, the inequality y ≥ 7 means that y can be any number greater than 7, or it can be equal to 7 (7, 7.001, 8, 9, 200, etc.). That little line underneath an inequality symbol means "or equal to."
How do we remember which one is which? "Less than" and "greater than" are easy to mix up, so we like to think of them as an incomplete Pac-Man (or, if you prefer, Ms. Pac-Man). Pac-Man, being the hungry circle he is, always wants to eat the bigger number, so his "mouth" will always be open towards the larger number.
How to Graph Inequalities
We can graph inequalities on a number line to get a better idea of how they're behaving. Just follow these steps.
- Find the number on the other side of the inequality sign from the variable (like the 4 in x > 4).
- Sketch a number line and draw an open circle around that number.
- Fill in the circle if and only if the variable can also equal that number.
- Shade all numbers the variable can be.
Here's what y ≤ 2 looks like:
Here's what y < 2 looks like:
Notice the subtle difference between the two graphs. In the first graph, the circle around the 2 is colored in. This is because y can be 2 in the first, but not the second.
Graphing Inequalities Examples
Example 1
j > -3.5
In this example, the circle around the -3.5 is not colored in and all numbers to the right of the circle are shaded. This is because -3.5 is less than j, or we could say that j is greater than -3.5.
Example 2
p ≠ ¾
Here the variable can be any number besides ¾, so we need to shade in everything that's not ¾.
Example 3
-10 ≥ x
The circle is colored in because x can be -10 and x can be smaller than -10, so we shade all numbers to the left.
Look Out: if you switch the terms on each side of the inequality, be very careful to change the sign, too. For example, x > 6 is the same as 6 < x.
Compound Inequalities
Compound inequalities are two or more inequalities combined in the same statement. They often include the words "and" or "or." With "and" inequalities, we only graph the numbers that satisfy both inequalities, a.k.a. the intersection of both inequalities. With "or" inequalities, we graph the numbers that satisfy either inequality, or both at the same time. In other words, we graph the combination, or union, of both inequalities.
Let's start by looking at an "or" example in depth.
y > -1 or y ≤ -3
If we break this apart, we can think of it as two separate inequalities:
y > -1
y ≤ -3
For an "or" inequality we combine all possible values of y onto one number line:
Now let's look at an "and" inequality:
-0.5 < z and z ≤ 0.25
For starters, we can combine forces and write the inequality like this:
-0.5 < z ≤ 0.25
Now tackle each side separately.
-0.5 < z
z ≤ 0.25
To finish up, we only graph the numbers that satisfy both conditions; i.e. the numbers greater than -0.5 and less than or equal to 0.25.