The point \((-3, 5)\) in the xy-plane is a solution to which of the following systems of inequalities?

1. TRANSLATE the point information

• Given information:
  • Point \((-3, 5)\) where \(-3\) is the x-coordinate and \(5\) is the y-coordinate
• What this tells us:
  • \(\mathrm{x = -3}\) (negative value)
  • \(\mathrm{y = 5}\) (positive value)

2. TRANSLATE coordinates into inequality conditions

• Since \(\mathrm{x = -3}\) and \(-3\) is negative: \(\mathrm{x \lt 0}\)
• Since \(\mathrm{y = 5}\) and \(5\) is positive: \(\mathrm{y \gt 0}\)
• Our point satisfies: \(\mathrm{x \lt 0}\) AND \(\mathrm{y \gt 0}\)

3. INFER the systematic checking approach

• We need to find which answer choice matches both conditions
• Each answer choice has two parts connected by "and" - both must be true

4. Check each answer choice systematically

• (A) \(\mathrm{x \gt 0}\) and \(\mathrm{y \gt 0}\): Does \(\mathrm{-3 \gt 0}\)? No. This fails immediately.
• (B) \(\mathrm{x \gt 0}\) and \(\mathrm{y \lt 0}\): Does \(\mathrm{-3 \gt 0}\)? No. This fails immediately.
• (C) \(\mathrm{x \lt 0}\) and \(\mathrm{y \gt 0}\): Does \(\mathrm{-3 \lt 0}\)? Yes. Does \(\mathrm{5 \gt 0}\)? Yes. Both true!
• (D) \(\mathrm{x \lt 0}\) and \(\mathrm{y \lt 0}\): Does \(\mathrm{-3 \lt 0}\)? Yes. Does \(\mathrm{5 \lt 0}\)? No. This fails.

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing the order of coordinates or misinterpreting inequality symbols

Students might think that because they see negative numbers, they need \(\mathrm{x \lt 0}\) and \(\mathrm{y \lt 0}\), not realizing that the y-coordinate \((5)\) is actually positive. Or they might confuse which coordinate is x and which is y.

This may lead them to select Choice D (\(\mathrm{x \lt 0}\) and \(\mathrm{y \lt 0}\)).

Second Most Common Error:

Incomplete INFER reasoning: Only checking one condition instead of both

Students might see that \(\mathrm{x \lt 0}\) is satisfied and stop there, not verifying that the y-condition also matches. They might pick the first choice where \(\mathrm{x \lt 0}\) is true without checking the y-condition.

This could lead to selecting Choice D or cause confusion between C and D.

The Bottom Line:

This problem requires careful attention to both the signs of coordinates AND the logical structure of compound inequalities with "and" - both conditions must be satisfied simultaneously.