What does the inequality y/x ≥ 2/3 - 6 represent?

What does the inequality y/x ≥ 2/3 - 6 represent?

• A. A line boundary dividing the plane
• B. A region on the graph where the ratio y/x is at least -16/3
• C. A constraint on x only
• D. A relationship indicating y is always greater than x

Answer: (B)

This question tests how to interpret y/x as something geometric in the plane. The expression 2/3 − 6 simplifies to −16/3, so the inequality is y/x ≥ −16/3. Here y/x is the slope of the line from the origin to the point (x, y). Requiring that slope to be at least −16/3 means every point whose line from the origin has slope at least that value belongs to the region. The boundary is the line through the origin with slope −16/3, which is y = (−16/3)x; the region includes that line. Note that x cannot be 0, since y/x is undefined there, so the y-axis isn’t part of the region, though points on the boundary line (except the origin, where the ratio isn’t defined) are included. So the description “a region on the graph where the ratio y/x is at least −16/3” captures the idea.

This question tests how to interpret y/x as something geometric in the plane. The expression 2/3 − 6 simplifies to −16/3, so the inequality is y/x ≥ −16/3. Here y/x is the slope of the line from the origin to the point (x, y). Requiring that slope to be at least −16/3 means every point whose line from the origin has slope at least that value belongs to the region. The boundary is the line through the origin with slope −16/3, which is y = (−16/3)x; the region includes that line. Note that x cannot be 0, since y/x is undefined there, so the y-axis isn’t part of the region, though points on the boundary line (except the origin, where the ratio isn’t defined) are included. So the description “a region on the graph where the ratio y/x is at least −16/3” captures the idea.