Which compound inequality could be represented by the graph? –4 ≤ x ≤ 4
–2 ≤ x ≤ –1
x ≤ –1 or x ≥ 0
x ≤ 3 or x ≥ –1

Many times, solutions lie between two quantities, rather than continuing endlessly in one direction. For example systolic (top number) blood pressure that is between 120 and 139 mm Hg is called borderline high blood pressure. This can be described using a compound inequality, b < 139 and b > 120. Other compound inequalities are joined by the word “or”.

When two inequalities are joined by the word and, the solution of the compound inequality occurs when both inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or, the solution of the compound inequality occurs when either of the inequalities is true. The solution is the combination, or union, of the two individual solutions.

Solving and Graphing Compound Inequalities in the Form of “or”

Let’s take a closer look at a compound inequality that uses or to combine two inequalities. For example, x > 6 or x < 2. The solution to this compound inequality is all the values of x in which x is either greater than 6 or x is less than 2. You can show this graphically by putting the graphs of each inequality together on the same number line.

The graph has an open circle on 6 and a blue arrow to the right and another open circle at 2 and a red arrow to the left. In fact, the only parts that are not a solution to this compound inequality are the points 2 and 6 and all the points in between these values on the number line. Everything else on the graph is a solution to this compound inequality.

Let’s look at another example of an or compound inequality, x > 3 or x ≤ 4. The graph of x > 3 has an open circle on 3 and a blue arrow drawn to the right to contain all the numbers greater than 3.

The graph of x ≤ 4 has a closed circle at 4 and a red arrow to the left to contain all the numbers less than 4.

What do you notice about the graph that combines these two inequalities?

Since this compound inequality is an or statement, it includes all of the numbers in each of the solutions, which in this case is all the numbers on the number line. (The region of the line greater than 3 and less than or equal to 4 is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality x > 3 or x ≤ 4 is the set of all real numbers!

You may need to solve one or more of the inequalities before determining the solution to the compound inequality, as in the example below.

Example

Problem

Solve for x.

3x – 1 < 8 or x – 5 > 0

Solve each inequality by isolating the variable.