Answer:
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.
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!!!