Section 7.4

Solving Compound Inequalities

What you should learn

To solve problems by making a diagram

To solve compound inequalities and graph their solution sets

To solve problems that involve compound inequalities

NCTM Curriculm Standards 2, 6 - 10

Introduction: The largest fish that spends its whole life in fresh water is the rare Pla beuk, found in the Mekong River in China, Laos, Cambodia, and Thailand. The largest specimen was reportedly 9 feet 10 1/4 inches long and weighed 533.5 pounds.

Such rare fish are sometimes displayed in aquariums. Aquariums can house freshwater or marine life and must be closely monitored to maintain the correct temperature and pH for the animals to survive. pH is a measure of acidity. To determine pH, a scale with values from 0 to 14 is used. One such scale is shown in the diagram below.

If we let p represent the value of the pH scale, we can express the different pH levels by using inequalities. For example, an acid solution will have a pH level of p0 and p < 7. When considered together, these two inequalities form a compound inequality. This compound inequality can also be written without using and in two ways.

0p < 7 or 7 > p 0

The statement 0p < 7 can be read o is less than or equal to p, which is less than 7. The statement 7 > p 0 can be read 7 is greater than p, which is greater than or equal to 0.

The pH levels of bases could be written as follows.

p > 7 and p14 or 7 < p 14 or 14p > 7

You can draw a diagram to help solve many problems. Sometimes a picture will help you decide how to work the problem. Other times the picture will show you the answer to the problem.

Exercise 1: On May 6, 1994, President Francoiis Mitterrand of France and Queen Elizabeth II of England officially opened the Channel Tunnel connecting England and France. After the ceremonies, a group of 36 English and French government officials had dinner at a restaurant in Calis, France, to celebrate the occasion. Suppose the restaurant staff used small tables that seat four people each, placed end to end, to form one long table. How many tables were needed to seat everyone?

Draw a diagram to represent the tables placed end to end. Use Xs to indicate where people are sitting. Let's start with a guess, say 10 tables.

Ten tables will seat 22 people. If we use an extra table, we can seat 2 more people. Now, let's look for a pattern.

 Number of Tables 10 11 12 13 14 15 16 17 Number of people seated 22 24 26 28 30 32 34 36

This pattern shows that the restaurant needed 17 tables to seat all 36 officials.

A compound inequality containing and is true only if both inequalities are true. Thus, the graphs of a compound inequality containing and is the intersection of the graphs of the two inequalities. The intersection can be found by graphing the two inequalities and then determining where these graphs overlap. In other words, draw a diagram to solve the inequality.

Exercise 2: Graph the solution set of x-2 and x < 5.

The solution set, shown in the bottom graph, ix {x|-2x < 5}. Note that the graph of x-2 includes the point -2. The graph of x < 5 does not include 5.

Exercise 3: Solve -1 < x + 3 < 5. Then graph the solution set.

The following example shows how you can solve a problem by using geometry, a diagram, and a compound inequality.

Exercise 4: Mai and Luis hope someday to compete in the Olympics in pairs ice skating. Each day they travel from their homes to an ice rink to practice before going to school. Luis lives 17 miles from the rink, and Mai lives 20 miles from it. If this were all the information given, determine how far apart Mai and Luis live.

Another type of compound inequality contains the word or instead of and. A compound inequality containing or is true if one of more of the inequalities is true. The graph of a compound inequality containing or is the union of the graphs of the two inequalities.

Exercise 5: Graph the solution set of x-1 or x < -4.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 19 - 57 odd, 59 - 68

Alternative Homework: Enriched: 18 - 52 even, 53 - 68

Extra Practice: Students book page 772 Lesson 7-4