- The best choice depends on the system of inequalities you write. I'd choose A.
- D
1. Let x and y represent the number of pencils and pens, respectively. Then the problem statement tells us ...
... x + y ≤ 25
... x ≥ y - 3²
We also know that the numbers of pens and pencils cannot be negative, so our system of inequalities can include ...
... x ≥ 0
... y ≥ 0
The attached graph shows the solution space as the purple area where the red and blue areas overlap. Because of the requirements we have imposed that x and y be non-negative, the solution space does not include any negative values, rendering answer choices B, C, and D irrelevant. Thus answer choice A is the only viable option for this system of inequalities.
It should be noted that the graph includes real numbers, not just integers. Viable solutions are actually restricted to integers, so one could argue that the entire solution space is not viable for that reason—not for the reasons given in B, C, and D.
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<em>Comment on alternatives</em>
If you don't include in the system of inequalities the restrictions x ≥ 0 and y ≥ 0, then the solution space does include negative numbers of pencils and pens. Now, you get into semantics.
B says "all solutions are not viable", which is clearly false.
C says "none of the solution region is viable", which is clearly false (and the same as B in regard to the number of viable solutions).
D says "not all solutions are viable", which is true regardless of the system of equations you choose to write (see comment on real numbers). So D may be the best alternate choice, if your system of inequalities allows negative values for pens.
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2. The appropriate translation of the verbiage given is that shown in answer choice D. It shows the cube of a sum divided by a product that includes a difference as a factor.
