Identifying Constraints in Inequalities

2.8.2 Identifying Constraints in Inequalities

2.8.2 • Identifying Constraints in Inequalities

Activity

Seniors in a high school student council are trying to develop a budget for the Senior Dance. The revenue from ticket sales needs to cover the cost of the meals and entertainment and also make a profit of at least $200 to be contributed to the school.

Here is some information they have gathered:

Last year, 120 people attended. It was a success and is expected to be even bigger this year. Up to 200 people will attend.
There needs to be at least 1 chaperone for every 20 students.
The ticket price cannot exceed $20 per person.

Here are some inequalities the seniors wrote about the situation. Each letter stands for one quantity in the situation. Determine what is meant by each letter in 1 – 4.

$t \leq 20$

Compare your answer:

$t \leq 20$ means that the ticket price, $t$ , must be less than or equal to $20.

$120 \leq p \leq 200$

Compare your answer:

$120 \leq p \leq 200$ means that between 120 and 200 people are expected to attend (120 is less than or equal to the number of people; the number of people is less than or equal to 200). $p$ represents the number of people attending the dance.

$p t - m \geq 200$

Compare your answer:

$p t - m \geq 200$ means that the total amount collected from ticket sales in dollars, $p t$ , minus the cost of meals and entertainment in dollars, $m$ , must be greater than or equal to $200.

$c \geq \frac{p}{20}$

Compare your answer:

$c \geq \frac{p}{20}$ means that the number of chaperones, $c$ , must be at least $\frac{1}{20}$ of the number of students attending.

Are you ready for more?

Extending Your Thinking

Kiran says we should add the constraint $t \geq 0$ .

What is the reasoning behind this constraint?

Compare your answer:

We can't have a negative ticket price.

What other “natural constraint” like this should be added?

Compare your answer:

By the same reasoning, we might want to include $m \geq 0$ . Note that the other variables should also be greater than or equal to 0, but those conditions are already implied by the existing constraints.

Self Check

A zookeeper feeds the tigers a meal of rabbits and chickens. The average rabbit weighs 4 pounds, and the average chicken weighs 5 pounds. Each tiger must eat at least 15 pounds of meat per day.The zookeeper writes the inequality $4 x + 5 y \geq 15$ to represent a tiger’s daily diet. What is meant by the letter $y$ in the inequality above?

Additional Resources

Translate to an Inequality

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like “more than” and “less than.” But others are not as obvious. The table shows some common phrases that indicate inequalities.

$>$	$\geq$	$<$	$\leq$
is greater than is more than is larger than exceeds	is greater than or equal to is at least is no less than is the minimum	is less than is smaller than has fewer than is lower than	is less than or equal to is at most is no more than is the maximum

Example

Translate each phrase to an inequality.

Twenty-seven less than $x \underset{⏟}{i s a t l e a s t}$ 48 $x - 17 \geq 48$

"is at least" means $\geq$

5 more than $z \underset{⏟}{i s l o w e r t h a n}$ 52 $z + 5 < 52$

"is at least" means $\geq$

Try it

Translate to an Inequality

Translate the following sentences into inequalities.

Nineteen less than $p$ is no less than 47.

Compare your answer:

Solution $p - 19 \geq 47$

Four more than $a$ is at most 15.

Compare your answer:

Solution $a + 4 \leq 15$