To decorate for a party, Rob and Loretta were each given a bag of balloons. There were 40 balloons between both bags. Rob used 8 balloons from his bag, and Loretta used 8 balloons from hers. The product of the number of balloons left in each bag is no more than 44. Let x represent the number of balloons that Rob had in his bag before he used any. Which choice represents the number of balloons left in their bags?

If Rob originally had more balloons than Loretta, what is the least number of balloons that Rob could have had before using any?

1) Option D is correct.

The choice that represents the number of balloons in their bags is x² - 40x + 300 ≥ 0

2) Option A is correct

If Rob had more balloons than Loretta, the least number of balloons that Rob van have before using any = 30.

Step-by-step explanation:

Complete Question

To decorate for a party, Rob and Loretta were each given a bag of balloons. There were 40 balloons between both bags. Rob used 8 balloons from his bag, and Loretta used 8 balloons from hers. The product of the number of balloons left in each bag is no more than 44.

Let x represent the number of balloons that Rob had in his bag before he used any. Which choice represents the number of balloons left in their bags?

A. x² - 40x - 300 = 0

B. x² - 40x - 300 < 0

C. x² - 40x = 300

D. x² - 40x + 300 ≥ 0

If Rob originally had more balloons than Loretta, what is the least number of balloons that Rob could have had before using any?

A.30

B.20

C.10

Solution

If x is the number of balloons that Rob had at the start, the number of balloons that Loretta would have at the start = (40 - x)

Then, they both use 8 balloons each for their designs,

The number of balloons Rob has after the design = (x - 8)

The number of balloons Lorretta has after the design = (40 - x - 8) = (32 - x)

The product of the balloons left isn't more than 44

(x - 8) × (32 - x) ≤ 44

32x - x² - 256 + 8x ≤ 44

44 ≥ -x² + 32x + 8x - 256

44 ≥ -x² + 40x - 256

44 + x² - 40x + 256 ≥ 0

x² - 40x + 300 ≥ 0

To solve the second part of the question, we just solve the expression obtained in the first part.

x² - 40x + 300 ≥ 0

Simplifying the left hand side

(x - 30)(x - 10) ≥ 0

The solutions of the inequality is

x ≤ 10 or x ≥ 30

Hence, if Rob had more balloons than Loretta, the feasible solution for x would be x ≥ 30 and the least number of balloons that Rob van have before using any = 30.

Hope this Helps!!!