Understanding the Meaning of an Inequality

2.9.3 Understanding the Meaning of an Inequality

2.9.3 • Understanding the Meaning of an Inequality

Activity

For questions 1 – 3, use the information about the orchards to answer each question.

A teacher is choosing between two options for a class field trip to an orchard. At each orchard, the same price applies to both chaperones and students.

At Orchard A, admission costs $9 per person, and 3 chaperones are required.

At Orchard B, the cost is $10 per person, but only 1 chaperone is required.

Which orchard would be cheaper to visit if the class has 8 students?

Orchard B.

Which orchard would be cheaper to visit if the class has 12 students?

Orchard B.

Which orchard would be cheaper to visit if the class has 30 students?

Orchard A.

Use the following information to answer questions 4 – 7.

To help her compare the cost of her two options, the teacher first writes the equation $9 (n + 3) = 10 (n + 1)$ , and then she writes the inequality $9 (n + 3) < 10 (n + 1)$ .

What does $n$ represent in each statement?

Compare your answer:

$n$ represents the number of students on the field trip.

In this situation, what does the equation $9 (n + 3) = 10 (n + 1)$ mean?

Compare your answer:

$9 (n + 3)$ is the cost of going to Orchard A, and $10 (n + 1)$ is the cost of going to Orchard B. The equation represents the two options costing the same amount.

What does the solution to the inequality $9 (n + 3) < 10 (n + 1)$ tell us?

Compare your answer:

The solution represents the number of students on the field trip at which it would be cheaper to visit Orchard A.

The teacher needs a visual aid to show the school budgeting committee. Graph the solution set to the inequality on the number line. Be prepared to show or explain your reasoning.

Compare your answer:

$n > 17$

A number line is shown. There is an open circle on 17. The number line to the right of 17 is highlighted.

Video: Understanding the Meaning of an Inequality

Watch the following video to learn more about the meaning of an inequality.

Access multimedia content

Self Check

A water park charges $17 per entry plus a $40 fee for raft rentals. An amusement park charges a $21 entry fee.

For the number of entries,

e

, which inequality represents the total cost of the water park being less than the total cost of the amusement park?

$17e\;+\;40\;>\;21e$
$17e\;<\;21e\;+\;40$
$17e\;+\;40\;<\;21e$
$17e\; - \;40 \;< \;21e$

Additional Resources

Translating Situations into Inequalities

Many real-life situations require us to use inequalities. Translating the details of these situations into inequalities is the first step in understanding them. Let’s look at how we can translate these situations into mathematical sentences.

Example 1

Imani won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. She can only spend up to the amount of the mini-grant.

Choose a variable to represent the quantity of tablet computers:

Let $n$ = the number of tablets.

Translate: Write a sentence that gives the information provided.

$254.12 times the number of tablets is no more than $4,000.

Translate into an inequality:

$254.12 n \leq 4000$

Imani’s situation is represented by the inequality $254.12 n \leq 4000$ .

Example 2

Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. Her monthly bill can be no more than $50.

Choose a variable to represent the quantity of text messages:

Let $t$ = the number of text messages.

Translate: Write a sentence that gives the information provided.

$28.80 plus $0.20 times the number of text messages is less than or equal to $50.

Translate into an inequality:

$28.80 + 0.20 t \leq 50$

Taleisha’s situation is represented by the inequality $28.80 + 0.20 t \leq 50$ .

Try it

Translating Situations into Inequalities

Read each problem below and write an inequality to represent the situation.

Angie has at most $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. She will also buy one bag of chips for $3.99.

Compare your answer:

$2.63 j + 3.99 \leq 20$

Jose wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. He must also pay $5 for parking. His budget for the party is less than $500.

Compare your answer:

$42.75 p + 5 < 500$