Lesson 5 May I Have More, Please? Solidify Understanding

Jump Start

Determine which inequality matches each phrase.

$x > 7$
$x \geq 7$
$x < 7$
$x \leq 7$

a.

No more than $7$

b.

A minimum of $7$

c.

More than $7$

d.

No less than $7$

e.

At least $7$

f.

A maximum of $7$

g.

Cannot exceed $7$

h.

Fewer than $7$

Learning Focus

Write and solve inequalities to model real situations.

Write solutions to inequalities using set builder and interval notation.

How can inequalities be used to find solutions to real problem situations?

Technology guidance for today’s lesson:

Basic Calculations — Square Root: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Exponents: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Add, Subtract, Multiply, and Divide: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Add and Subtract: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Elvira, the cafeteria manager, needs to be careful with her spending, and she manages the cafeteria budget so that they can serve the best food at the lowest cost. To do this, Elvira keeps good records and analyzes all her budgets.

1.

Elvira’s cafeteria has those cute little cartons of milk that are typical of school lunch. The milk supplier charges $$ 0.35$ per carton of milk, in addition to a delivery charge of $$ 75$ . What is the maximum number of milk cartons that Elvira can buy if she has budgeted $$ 500$ for milk?

a.

Write and solve an inequality that models this situation.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

2.

Students love to put ranch dressing on everything, so Elvira needs to keep plenty in stock. The students eat about $2.25$ gallons of ranch each day! Elvira started the school year with $130$ gallons of ranch dressing. She needs to have at least $20$ gallons left when she reorders to have enough in stock until the new order comes. For how many days will her ranch dressing supply last before she needs to reorder?

a.

Write and solve an inequality that models this situation.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

3.

The prices on many of the cafeteria foods change during the year. Elvira finds that she has ordered veggie burgers four times and paid $$ 78$ , $$ 72$ , $$ 87$ , and $$ 90$ on the orders. To stay within her budget, Elvira needs to be sure that the average order of veggie burgers is not more than $$ 82$ . How much can she spend on the fifth order to keep the average order within her budget?

a.

Write and solve an inequality that models this situation.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

4.

Elvira can purchase ready-made pizzas for $$ 14.50$ each. If she makes them in the cafeteria, she can spend $$ 44.20$ on ingredients and $$ 6.25$ per pizza on labor. For how many pizzas is it cheaper for the cafeteria to make the pizzas themselves rather than buy them ready-made?

a.

Write and solve an inequality that models this situation.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

5.

Elvira is comparing prices between two different suppliers of fresh lettuce. Val’s Veggies charges $$ 250$ for delivery plus $$ 1.50$ per bag of lettuce. Sally’s Salads charges $$ 100$ for delivery plus $$ 4.00$ per bag of lettuce. How many bags of lettuce must be purchased for Val’s Veggies to be the cheaper option?

a.

Write and solve an inequality that models this situation.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

6.

Each student who buys a school lunch pays $$ 2.75$ . The cafeteria typically brings in between $$ 1, 168.75$ and $$ 1, 438.25$ each day. How many students does the cafeteria usually serve in a day?

a.

Model this situation using an inequality.

b.

Describe in words the quantities that would work in this situation.

c.

Write your answer in both interval and set notation.

Ready for More?

River A and River B combine together to form River C. The flow rate of the River C is less than or equal to the sum of the flow rates of the Rivers A and B, but it is greater than or equal to either of the individual flow rates of Rivers A and B. One of the three rivers flows at a rate of $52 gallons per minute$ . Another river flows at a rate of $38 gallons per minute$ .

a.

Find the minimum flow rate, $f$ , of the third river.

b.

Find the maximum flow rate, $f$ , of the third river.

c.

Write an inequality that describes the possible flow rates, $f$ , of the third river.

Takeaways

Using properties of inequalities to solve inequalities:

Adding Notation, Vocabulary, and Conventions

Compound inequality:

Examples:

Vocabulary

compound inequality in one variable
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we wrote inequalities to model contexts that had a range of solutions. We used the properties of inequalities to solve the inequalities and found that solving inequalities is very similar to solving equations, but we must be careful when multiplying or dividing by a negative number to reverse the inequality sign. We used interval and set notation to write solutions and learned that many of the solutions that we write are compound inequalities.

Retrieval

1.

Graph each equation and find the point where they intersect.

$y = - 5 x + 3$

$y = 3 x - 5$

Point of intersection: