Lesson 3 Too Big or Not Too Big, That Is the Question Solidify Understanding

Jump Start

1.

Find at least 5 coordinate pairs $(x, y)$ that make the following inequality true:

$2 x + 4 y < 20$

Represent each of the following statements algebraically using inequality symbols and variables to represent the quantities described in the problems: (Remember to define the meaning of the variables you use in your inequalities.)

2.

It will cost at least $$ 50$ to purchase two different video games for the evening activity.

3.

We expect that at most $60$ children and adults will attend the family reunion.

4.

No more than $30$ minutes should be spent preparing breakfast and lunch.

5.

The total cost for a morning and afternoon snack should not exceed $$ 10$ .

Learning Focus

Graph the solution set for linear inequalities in two variables.

How can I find the complete set of points that satisfy a given constraint?

How do I represent the complete solution set?

Technology guidance for today’s lesson:

Basic Graphing of Linear Inequalities: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

As Carlos is considering the amount of money available for purchasing cat pens and dog runs he realizes that his father’s suggestion of boarding “the same number of each, perhaps $12$ cats and $12$ dogs” may not be reasonable. Why?

Start-up costs: Carlos and Clarita plan to invest much of the $$ 1, 280$ they earned from their last business venture to purchase cat pens and dog runs. It will cost $$ 32$ for each cat pen and $$ 80$ for each dog run.

1.

Find at least 5 more combinations of cats and dogs that would be “too big” based on this start-up cost constraint. Plot each of these combinations as points on a coordinate grid using the same color for each point.
Find at least 5 combinations of cats and dogs that would be “not too big” based on this start-up cost constraint. Plot each of these combinations as points on a coordinate grid using a different color for the points than you used in 1a.
Find at least 5 combinations of cats and dogs that would be “just right” based on this start-up cost constraint. That is, find combinations of cat pens and dog runs that would cost exactly $$ 1, 280$ . Plot each of these combinations as points on a coordinate grid using a third color.

2.

What do you notice about these three different collections of points?

3.

Write an equation for the line that passes through the points representing combinations of cat pens and dog runs that cost exactly $$1,280$ . What does the slope of this line represent?

Carlos and Clarita don’t have to spend all of their money on cat pens and dog runs, unless it will help them maximize their profit.

4.

Shade all of the points on your coordinate grid that satisfy the start-up costs constraint.

5.

Write a mathematical rule to represent the points shaded in problem 4. That is, write an inequality whose solution set is the collection of points that satisfy the start-up costs constraint.

In addition to start-up costs, Carlos needs to consider how much space he has available, based on the following:

Space: Cat pens will require $6 {ft}^{2}$ of space, while dog runs require $24 {ft}^{2}$ . Carlos and Clarita have up to $360 {ft}^{2}$ available in the storage shed for pens and runs, while still leaving enough room to move around the cages.

6.

Write an inequality to represent the solution set for the space constraint. Shade the solution set for this inequality on a different coordinate grid.

What do you think? What recommendation would you give to Carlos and Clarita regarding how many cats and dogs to plan on boarding, and what argument would you use to convince them that your recommendation is reasonable?

Ready for More?

You can come up with the equation of a boundary line using many different strategies. Try to write the equation of the boundary line for the space constraint in each of the following ways: (You may have to find some additional information to try each strategy.)

a.

Directly from the words used to describe the constraint.

b.

Using just the $x$ - and $y$ -intercepts.

c.

Using any two points you might have found on the boundary line.

d.

Using the “exchange rate” between cats and dogs.

Takeaways

Linear equations can be written in the following forms:

Standard form:

Slope-intercept form:

Point-slope form:

The solution set to a linear inequality is a half plane, that is, .

The solution set to an inequality constraint for a context like Pet Sitters is .

The boundary line for a linear constraint may be written in form to reveal , or form to reveal .

Vocabulary

half-plane
linear function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to find and represent all of the points in the solution set for a linear inequality in two variables.

Retrieval

1.

Identify which of the given points are solutions to the following linear equation. Select all that are solutions.

$y = - 4 x + 5$

A.

$(- 4, 5)$

B.

$(- 2, 13)$

C.

$(0, - 4)$

D.

$(2, - 3)$

E.

$(4, - 11)$

2.

Find a missing value that will make each ordered pair be a solution to the given equation.

$2 x + y = 12$

$(0, \underset{―}{})$
$(\underset{―}{}, 0)$
$(- 1, \underset{―}{})$
$(5, \underset{―}{})$
$(\underset{―}{}, 6)$

3.

Graph the solution set for this inequality on the number line: $5 (x - 3) + 2 \leq 2 (x - 5)$