 Concrete flatwork at the Logan Water Tank (Logan, UT) by ABCO construction. |
Modeling Problem Situations with Inequalities Objective: When done with this lesson, you will have demonstrated how to model and solve problem situations with inequalities. Washington State GLE: 1.5.2
Approximate completion time: 3 hours |
Do you think that it could be too hot for a company to mix and then "pour concrete" for a job?
Do you think that it could be too cold?
Engineers, manufacturers, and contractors work hard to know what the ideal temperature would be for working with concrete. They also know that there is a range of temperatures that would work well. Depending on the concrete mix, the type of job, the humidity in the air, the wind speed, and other variables a company might use any one of the following "inequalities" as a guide:
40 ≤ T ≤ 80 would mean that the air temperature must be between 40° and 80° Fahrenheit including 40° and 80°.
Each of these are called inequalities. An equality would be used if there were only one temperature that would work. Yet since many temperatures are allowed, an inequality works best to describe what is possible. The graphs of inequalities demonstrate this quite well.
All the T values between and including 40 to 80. All the T values between and including 50 to 70. All the T values between and including 40 to 90.
More than concrete or temperatures, we use inequalities for all sorts of situations. Can you think of a situation that uses an inequality? Try to guess which of the following situations use inequalities. (Click on the link for the picture.)
If you said all of those situations are about inequalities, then you were correct!
Let's look now at an example of using inequalities to solve a problem situation.
Problem: Joe wants to load his truck with bags of cement. He knows his truck has a maximum payload of 1,550 pounds. He already has 400 pounds of sand and gravel in his truck. Knowing he doesn't want to go over 1,550 pounds, how many 80 pound bags of cement could he carry?
Solution: Using an inequality to model this situation and algebra to solve, we have the following:
Let b = the # of cement bags he could carry You must define your variable. 400 + 80b ≤ 1550
The total of 400 pounds of sand/gravel plus the 80 pounds per bag must not exceed 1550 pounds. 80b ≤ 1150 Subtracting 400 from both sides. b ≤ 1150 ÷ 80 Dividing both sides by 80. b ≤ 14.375 Simplifying. Since b is the number of cement bags he could carry, the most he could carry would be 14.375 bags. However, Joe would probably not be able to buy (nor want to buy) .375 of a bag. Therefore Joe can carry up to 14 bags of cement and still be under his maximum payload for his truck.
Note that Joe may likely need much less than 14 bags of cement. This inequality allows him to have less than or equal to 14 bags.
After looking at the algebra in that last example, many people would think that the rules of algebra for solving inequalities are the same as they are for solving equations.
BE CAREFUL!!! There are some things that are unique to solving inequalities.
INVESTIGATION:
Copy and complete the table by writing the appropriate inequality.
Remember that "greater than" means farther to the right on a number line.
Make a conclusion about what you see in this investigation by finishing this statement:
"When you take the opposite of numbers, then the inequality sign ...."
In general, the only new thing to consider when solving an inequality is this rule:
Look at this last example before you start your practice.
Problem: Solve the inequality -2(3 + 4x) < 46 .
Also include a graph of the solution.Solution:
Note the switch in direction of the inequality symbol.Therefore, x > 5 is the solution and here is the graph of that.
(Do you notice that there is an open circle at 5? That means that the value of 5 is not allowed.)
PRACTICE: For some practice on solving inequalities, be sure to use the SAS inSchool activity #962. (Instructions found here)
For some more practice, download and complete these practice problems.
