15 Compound Inequalities and Interval Notation PDF

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Math 043

Compound Inequalities and Interval Notation

Compound Inequalities and Interval Notation In previous courses, you should have covered how to solve linear inequalities. In this section we learn how to solve compound inequalities. Before we do, however, we need to discuss the different notations that may be used to show the solution to an inequality. Solution sets can be described using inequalities, number lines, or intervals. The following chart illustrates the different forms that can be used to describe the solution set.

Example 1: Graph the following inequalities on a number line, then write the corresponding interval(s). a) x>−2 Interval Notation ____________________________

b)

x≤3

Interval Notation ____________________________

Math 043

Compound Inequalities and Interval Notation

c)

−2≤ x −2∨x ≤ 3

Interval Notation ____________________________

The difference between “and” and “or” is very important. If a compound inequality has the word “and” between the two inequalities, a number must satisfy BOTH inequalities at the same time to be part of the solution set. For example, the solution to x< 5∧x >−2 would only include the numbers that are both less than 5 and at the same time greater than -2. On a number line, this would look like:

In interval notation:

Math 043

Compound Inequalities and Interval Notation

If a compound inequality has the word “or” between the two inequalities, a number need only satisfy EITHER of the inequalities to be part of the solution set. For example, the solution to x←3∨x >2 would include any number that is less than -3. It would also include any number greater than 2. On a number line, this would look like:

In interval notation:

To solve a compound inequality, solve each of the inequalities separately, then determine the solution set of the combined inequalities making sure to use the “and” or “or” that is given. Example 2:

Solve each of the compound inequalities and write your answer in interval notation.

a)

2 x +1 ≤5∧−3 x> 9

b)

4 −2 x ≤−2∨3 x+ 5 ≤ 8

Math 043

Compound Inequalities and Interval Notation

c)

5 x+2 8

d)

5 x+2 8

You could also be given a “three-part inequality.” It is written as one statement with two inequality symbols in it. To solve a three part inequality, your goal is to isolate the variable in the middle of the two inequality symbols. You do this by adding, subtracting, multiplying, and/or dividing ALL THREE PARTS of the inequality by the same number. Recall that multiplying or dividing an inequality by a NEGATIVE number reverses the direction of the inequality symbol. Example 3: a)

Solve each of the inequalities and write your answer in interval notation. −3 ≤2 x+ 5≤ 12

Math 043

Compound Inequalities and Interval Notation

2 x +5 ≤3 3

b)

−1≤

c)

−8 ≤2 −5 x ≤ 27...