# The concept of compound inequalities

### How to solve compound inequalities with variables on all sides

In other words, both statements must be true at the same time. Think about that one for a minute. You may not see the word "and" in the original inequality, but when you read the inequality aloud, you will say the word "and". When the two inequalities are joined by the word or, the solution of the compound inequality occurs when either of the inequalities is true. Example Solve for x. Do this by adding, subtracting, multiplying, and dividing the same thing to all three parts of the inequality at the same time. The following video presents two examples of how to draw inequalities involving AND, as well as write the corresponding intervals. In this section you will see that some inequalities need to be simplified before their solution can be written or graphed. In this case, the solution is all the numbers on the number line. Sometimes it helps to draw the graph first before writing the solution using interval notation. Graphing Linear Inequalities You can define a range of values using a single inequality statement, called a compound inequality.

Notice that you should always write the lower boundary on the left and the upper boundary on the right. You may not see the word "and" in the original inequality, but when you read the inequality aloud, you will say the word "and".

In a sense we are uniting these two answers in order to include both. Graphing Linear Inequalities You can define a range of values using a single inequality statement, called a compound inequality. When you place both of these inequalities on a graph, we can see that they share no numbers in common. It is the overlap, or intersection, of the solutions for each inequality. In this section you will see that some inequalities need to be simplified before their solution can be written or graphed. Everything else on the graph is a solution to this compound inequality. Figure 7.

### Triple inequalities

Notice that this is a bounded inequality. Everything else on the graph is a solution to this compound inequality. This is where both of these statements are true at the same time. It describes a range or interval of numbers, whose lower boundary is a, and whose upper boundary is b. The solution to an and compound inequality are all the solutions that the two inequalities have in common. 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. Can you see why we need to write them as two separate intervals? For example systolic top number blood pressure that is between and mm Hg is called borderline high blood pressure. In this section you will see that some inequalities need to be simplified before their solution can be written or graphed. 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.
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