What you should learn

To solve inequalities by using addition and subtraction

NCTM Curriculm Standards 2, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Set-builder Notation

Introduction: I n 1990, the US Department of Agriculture issued new dietary guidelines. These guidelines recommend that people greatly reduce their fat intake. You recommended calorie intake depends on your height, desired weight, and physical activity. The average 14-year-old is 5 feet 2 inches tall and weights 107 pounds. Boys should consume about 2434 Calories per day and girls 2208 Calories per day to maintain this weight.

Oliana learned in health class that no more than 30% of her calorie intake shoud come from fat. For her 2030-Calorie-a-day diet, that means no more than 68 grams of fat. She keeps track of her snacks for one day and records their fat content, as shown in the table above. How many grams of fat can Oliana have in the other foods she eats that day to stay within the guidelines?

Let's write and inequality to represent the problem. Let g represent the remaining grams fo fat that Oliana can eat that day.

15 + 8 + 4 + 2 + g 68

That is, 29 + g 68.

The symbol indicates less than or equal to. It is used in this situation because the total number of grams of fat in Oliana's daily diet should be no greater than 68. If this were an equation, we would subtract 29 from (or add -29 to) each side. Can the same procedure be used in an inequality?

Let's explore what happens if inequalities are solved in the same manner as equations. We know that 7 > 2. What happens when you add or subtract the smae quantity to each side of the inequality? We can use number lines to model the situation.

Add 3 to each side.

7 > 2

7 + 3 > 2 + 3

10 > 5

Subtract 4 from each side.

7 > 2

7 - 4 > 2 - 4

3 > -2

In each case, the inequality holds true. These examples illustrate two properties of inequalities.

Addition and Subtraction Properties for Inequalities: For all numbers a, b, and c, the follwoing are true.

1. If a > b, then a + c > b + c and a - c > b - c

2. If a < b, then a + c < b + c amd a - c < b - c.

 

These properties are also true when > and < are replaced by and . So, we can use these properties to obtain a solution to the application at the beginning of the lesson.

Exercise 1: Refer to the application at the beginning of the lesson.

Solve 29 + g 68

29 - 29 + g68 - 29

g39

The solution set can be written as {all numbers less than or equal to 39}.

Now Check...

 

 

The solution to the inequality in Exercise 1 was expressed as a set. A more concise way of writing a solution set is to use set-builder notation. The solution in set-builder notation is {g|g39}. This is read the set of all nubmers g such that g is less than or equal to 39.

In lesson 2-4, your learned that you can show the solution to an inequality on a graph. The solution to exercise 1 is shown on the number line below.

The closed circle at 39 tells us that 39 is included in the inequality. The heavy arrow (colored pink) pointing to the left shows that it also includes all numbers less than 39.

Exercise 2: Solve 13 + 2z < 3z - 39. Then graph the solution.

Verbal problems containing phrases like greater than or less than can often be solved by using inequalities. The following chart shows some other phrases that indicate inequalities.

Inequalities
<	>
Less than Fewer than	Greater than More than	At most No more than Less than or equal to	At least No less than Greater than or equal to

Exercise 3: Alvaro, Chip, and Solomon have earned $500 to buy equipment for their bank. They have already spend $275 on a used guitar and a drum set. They are now considering buying a $125 amplifier. What is the most they can spend on promotional materials and T-shirts for the band if they buy the amplifier? (Use your problem solving skills)

When solving problems involving equations, it is often necessary to write an equation that represents the words in the problem. This is also true of inequalities.

Exercise 4: Write an inequality for the sentence below. Then solve the inequality and check the solution.

Three times a number is more than the difference of twice that number and three.

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: 17 - 49 odd, 50 - 58

Alternative Homework: Enriched: 18 - 44 even, 45 - 58

Extra Practice: Students book page 771 Lesson 7-1

Extra Practice Worksheet: Click Here.

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