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.

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.

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

** Extra Practice Worksheet:** Click Here.