Using Inequalities to Solve a Problem

2.10.2 Using Inequalities to Solve a Problem

2.10.2 • Using Inequalities to Solve a Problem

Activity

Use the following information to answer questions 1 – 2.

Han is about to mow some lawns in his neighborhood. His lawn mower has a 5-gallon fuel tank, but Han is not sure how much gasoline is in the tank.

He knows, however, that the lawn mower uses 0.4 gallon of gasoline per hour of mowing.

What are all the possible values for $x$ , the number of hours Han can mow without refilling the lawn mower?

Compare your answer:

$x \leq 12.5,$ and $x \geq 0$ .

Write one or more inequalities to represent your response. Be prepared to show your reasoning.

Compare your answer:

$x \leq \frac{5}{0.4}$ or $x \leq 12.5$ and $x \geq 0$ . If the tank is completely full, Han can mow for at most 12.5 hours without refilling. If it is all the way empty, he can mow for 0 hours. It is not possible for Han to mow for a negative number of hours.

Video: Using Inequalities to Solve a Problem

Watch the following video to learn more about how to solve a problem using inequalities.

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Self Check

Samantha is feeding the horses at her barn. She has 6 buckets of oats. If each horse eats

\frac34

bucket of oats, how many horses can she feed at most?

$x\leq10$ , Samantha can feed a maximum of 10 horses.
$x\leq8$ , Samantha can feed a maximum of 8 horses.
$x\leq4$ , Samantha can feed a maximum of 4 horses.
$x\leq6$ , Samantha can feed a maximum of 6 horses.

Additional Resources

Using Inequalities to Solve a Problem

Let’s look at an example where we might be able to solve a problem using inequalities.

A school is repainting its classroom walls. The school has at most 30 gallons of paint. The painter needs $\frac{5}{2}$ gallons of paint for each classroom. How many classrooms can the school paint at most?

We know that the answer is greater than or equal to 0 classrooms, so $x \geq 0$ .

We also know that each classroom requires $\frac{5}{2}$ or 2.5 gallons of paint.

Let’s take a guess at how many classrooms and check if we are close to the solution. How many gallons of paint would we need to paint 10 classrooms?

$10 \times 2.5 = 25$ , and since $25 < 30$ , it could be possible for the school to paint 10 classrooms.

We can try 11 classrooms.

$11 \times 2.5 = 27.5$ , and since $27.5 < 30$ , it could be possible for the school to paint 11 classrooms.

We can try 12 classrooms.

$12 \times 2.5 = 30$ , and since $30 = 30$ , the most classrooms the school could paint is 12 classrooms.

Try it

Using Inequalities to Solve a Problem

Use the following information for questions 1 – 2.

While working in a car factory, a worker can use a drill for as long as the batteries last. The worker has at most 9 fully charged batteries. Each car the worker helps assemble uses $\frac{3}{4}$ of a full battery.

What are all of the possible values for $c$ , the number of cars the work can help assemble? Write one or more inequalities to represent your response.

Compare your answer:

$c \leq 12$ and $c \geq 0$

Explain the meaning of the inequalities.

Compare your answer:

If the worker has 9 fully charged batteries, they can help assemble up to 12 cars. If the worker has no batteries, they can assemble 0 cars. It is not possible for the worker to assemble less than 0 cars.