Answered

Select the correct answer.

Which situation could be represented by this system of linear inequalities?

[tex]\[
\begin{aligned}
4x + 5y & \leq 180 \\
x + y & \geq 40
\end{aligned}
\][/tex]

A. At a pharmacy, packs of generic cotton balls are sold for \[tex]$4 each and packs of name brand cotton balls are sold for \$[/tex]5 each. At the start of the day, the pharmacy has 40 packs of cotton balls that will earn a minimum sales revenue of \$180.

B. A grocery store sells oranges in 4-pound and 5-pound bags. The store sells no more than 180 pounds of oranges in a day, and they want to sell at least 40 bags of oranges each day.

C. On a math test, questions are worth either 4 or 5 points each. The test has 40 questions in all with a maximum score of 180 points.

D. A bookstore sells sets of notebooks which contain either 4 or 5 notebooks each. The owner wants to sell at least 40 sets of notebooks each day, which is a minimum of 180 notebooks.



Answer :

GinnyAnswer
To find the correct scenario that matches the given system of linear inequalities, we need to carefully analyze the constraints imposed by each inequality:

1. \(4x + 5y \leq 180\): This inequality suggests a restriction where the combination of two variables, \(x\) and \(y\), multiplied by their respective coefficients must not exceed 180. This is a maximum constraint.
2. \(x + y \geq 40\): This inequality indicates that the sum of the two variables must be at least 40. This is a minimum constraint.

Let's analyze each option one by one:

Option A:
- Packs of cotton balls are sold for \[tex]$4 and \$[/tex]5 each.
- The pharmacy starts with 40 packs and will earn a minimum of \$180.
- This scenario implies a minimum sales revenue of \$180, which aligns with a constraint \(4x + 5y \geq 180\). However, our given inequality is \(4x + 5y \leq 180\), a maximum constraint. Therefore, this option is incorrect.

Option B:
- A grocery store sells oranges in 4-pound and 5-pound bags.
- The store sells no more than 180 pounds of oranges in a day, and they want to sell at least 40 bags of oranges each day.
- The constraints here directly match our inequalities: \(4x + 5y \leq 180\) (no more than 180 pounds) and \(x + y \geq 40\) (at least 40 bags). Thus, this option is correct.

Option C:
- A math test with questions worth either 4 or 5 points each.
- The test has 40 questions total with a maximum score of 180 points.
- This scenario implies a sum of the points or scores, where the maximum score constraint \(4x + 5y \leq 180\) fits. However, \(x + y \geq 40\) means at least 40 questions, which matches the situation well too. However, the given inequalities seem equally well addressed by another more fitting scenario. Let's compare to the others first.

Option D:
- A bookstore sells sets containing either 4 or 5 notebooks each.
- They wish to sell at least 40 sets, which is a minimum of 180 notebooks.
- If a minimum of 180 notebooks were needed, we would expect the constraint \(4x + 5y \geq 180\). Our constraint is \(4x + 5y \leq 180\), hence this option does not fit well.

Thus, after comparing all the options carefully:

Correct answer: B: A grocery store sells oranges in 4-pound and 5-pound bags. The store sells no more than 180 pounds of oranges in a day, and they want to sell at least 40 bags of oranges each day.