When we have variables with inequalities, we have to make sure that we don't make assumptions about the SIGN, i.e. whether it's positive or negative.
If gh < 0 < gk, is g < 0?
We definitely can't divide through by g to get h < 0 < k, because if g were negative, the signs would flip to h > 0 > k. There are 2 possible scenarios:
1. g is positive, h is negative, and k is positive
2. g is negative, h is positive, and k is negative
If we're wondering whether g is negative, we could rephrase the question as: is h > 0 > k ?
1) k < h
We know from the question stem that h & k must have opposite signs (one positive, one negative). If k is less than h, then k must be negative, so we have scenario #2: g is negative, h is positive, and k is negative.
This is sufficient to answer the question.
2) 0 < h
Again, we know from the question stem that h & k must have opposite signs (one positive, one negative). If h > 0, then h must be positive, so we have scenario #2: g is negative, h is positive, and k is negative.
This is sufficient to answer the question.
The answer is D.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
