Lesson 6 All for One, One for All Solidify Understanding

Jump Start

Here are all of the constraints Carlos and Clarita have identified for their pet sitting business.

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.
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.
Feeding time: Carlos and Clarita estimate that cats will require $6$ minutes twice a day—morning and evening—to feed and clean their litter boxes, for a total of $12$ minutes per day for each cat. Dogs will require $10$ minutes twice a day to feed and walk, for a total of $20$ minutes per day for each dog. Carlos can spend up to $8$ hours each day for the morning and evening feedings, but needs the middle of the day off for baseball practice and games.
Pampering time: The twins plan to spend $16$ minutes each day brushing and petting each cat, and $20$ minutes each day bathing or playing with each dog. Clarita needs time off in the morning for swim team and evening for her art class, but she can spend up to $8$ hours during the middle of the day to pamper and play with the pets.

Write the system of inequalities that model these constraints.

Learning Focus

Find the solution set for a system of linear inequalities.

How do I represent the solutions that satisfy all of the constraints being placed on a context?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita have found a way to represent combinations of cats and dogs that satisfy each of their individual Pet Sitters constraints, but they realize that they need to find combinations that satisfy all of the constraints simultaneously. Why?

1.

Begin by listing the system of inequalities you have written to represent the start-up costs and space Pet Sitters constraints.

2.

Find at least 5 combinations of cats and dogs that would satisfy both of the constraints represented by this system of inequalities. How do you know these combinations work?

3.

Find at least 5 combinations of cats and dogs that would satisfy one of the constraints, but not the other. For each combination, explain how you know it works for one of the inequalities, but not for the other?

4.

Shade a region on the coordinate grid that would represent the solution set for the system of inequalities. Explain how you found the region to shade.

Pause and Reflect

5.

Rewrite your systems of inequalities to include the additional constraints for feeding time and pampering time.

6.

Find at least five combinations of cats and dogs that would satisfy all of the constraints represented by this new system of inequalities. How do you know these combinations work?

7.

Find at least five combinations of cats and dogs that would satisfy some of the constraints, but not all of them. For each combination, explain how you know it works for some inequalities, but not for others?

8.

Shade a region of the coordinate grid that would represent the solution set to the system of inequalities consisting of all four Pet Sitters constraints. Explain how you found the region to shade.

9.

Shade a region in Quadrant I of the coordinate grid that would represent all possible combinations of cats and dogs that satisfy the four Pet Sitters constraints. This set of points is referred to as the feasible region since Carlos and Clarita can feasibly board any of the combinations of cats and dogs represented by the points in this region without exceeding any of their constraints on time, money, or space.

10.

How is the feasible region shaded in problem 9 different from the solution set to the system of inequalities shaded in problem 8?

Ready for More?

Graph the solution set for the following system of inequalities. To clearly define the solution set, you will also want to find the vertices of the polygon that represents the feasible region for the solutions of the system.

${\begin{matrix} 2 x + y \geq 4 \\ \frac{1}{2} x + y \leq 2 \\ x - y \leq 4 \end{matrix}$

Takeaways

To find the solution for a system of inequalities:

To define the feasible region for a system of constraints that models a context:

Adding Notation, Vocabulary, and Conventions

Given a system of inequalities ${\begin{matrix} A_{1} x + B_{1} y \leq C_{1} \\ A_{2} x + B_{2} y \geq C_{2} \\ A_{3} x + B_{3} y \leq C_{3} \end{matrix}$

The solution set for the system of inequalities is:

If the system represents the constraints in a modeling context, then the feasible region is:

Vocabulary

compound inequality in two variables
feasible region
solution set for the system of inequalities
solution to an equation (satisfies an equation)
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to find the solution to a system of inequalities by finding the points in the coordinate plane that satisfy all of the inequalities simultaneously. We also used the solution to the system of inequalities for the Pet Sitters constraints to find the feasible region of viable options for Carlos and Clarita to consider when starting their business.

Retrieval

Determine the value of the square root.

1.

$\sqrt{64}$

2.

$\sqrt{225}$

Solve the equation by inspection or guess and check.

3.

$x^{2} = 36$

4.

$4^{x} = 1,024$

5.

The boundary line for the inequality is provided on the coordinate grid. Shade the half plane that contains all solutions for the inequality.

$4 x - 9 y < 36$