Answer:
- None of the above. The offered choices answer a different question. (see below for an explanation)
- 30
Step-by-step explanation:
The problem statement tells us the number of balloons Rob originally had is represented by x. Since Rob and Loretta together have 40 balloons, the number Loretta started with is (40-x).
After each used 8 balloons, Rob has x-8 remaining, and Loretta has (40-x)-8 = 32-x remaining. The question asks "which choice represents the number of balloons left in their bags?"
We expect perhaps a list {x-8, 32-x} or maybe the sum x-8+32-x = 24 as representative of the number of balloons left in their bags. Instead, we're given a quadratic equation or inequality. An answer to the question is not present here.
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The problem statement also tells us that the product of the numbers of remaining balloons is at most 44. That is, ...
... (x-8)(32-x) ≤ 44
Multiplying this out gives ...
... -x² +40x -256 ≤ 44
If we add the opposite of the left side of the inequality, we get ...
... 0 ≤ x² -40x +300
Swapping sides gives ...
... x² -40x +300 ≥ 0 . . . . . . . matches the second offered choice
This inequality answers the question, "which choice is representative of the constraint on the product of the numbers balloons left in their bags?"
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The solution to this inequality can be found several ways. I like graphing or factoring (or both). A graph shows zeros at x=10 and x=30. (These zeros are part of the solution set.) These zeros mean the inequality can be factored as ...
... (x -10)(x -30) ≥ 0
We know the product will have a positive sign when both of the factors have the same sign—that is, for x < 10 or x > 30. This means the solution to the inequality is ...
... x ≤ 10 ∪ 30 ≤ x . . . . . . . Rob's starting number must be ≤10 or ≥30
Of course, when Rob's starting number of balloons is 30, Loretta's starting number will be 10, and vice versa. If Rob starts with more than Loretta, the least number he could start with is 30. (x ≥ 30)
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Comment on the problem
Math is supposed to teach critical thinking. Please have your teacher explain to you how the answer choices offered here are an answer to the question asked here. (I think your teacher may find that task difficult. Do not accept "the computer said so" for an explanation.)
