Question:y gt -4x + 15For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the problem requirements

• Given: The inequality \(\mathrm{y \gt -4x + 15}\)
• Need to find: Which table has ALL coordinate pairs that satisfy this inequality
• What this means: For each (x,y) pair, when we substitute the values, y must be strictly greater than -4x + 15

2. INFER the solution strategy

• We need to test every single pair in each table systematically
• If even one pair fails the inequality test, that entire table is eliminated
• Only the table where ALL pairs work is correct

3. SIMPLIFY by testing each table systematically

Table A Check:

• For (-2, 25): Is \(\mathrm{25 \gt -4(-2) + 15}\)?
  \(\mathrm{25 \gt 8 + 15}\)
  \(\mathrm{25 \gt 23}\)? ✓ True
• For (0, 16): Is \(\mathrm{16 \gt -4(0) + 15}\)?
  \(\mathrm{16 \gt 15}\)? ✓ True
• For (1, 10): Is \(\mathrm{10 \gt -4(1) + 15}\)?
  \(\mathrm{10 \gt -4 + 15}\)
  \(\mathrm{10 \gt 11}\)? ✗ False

Table A fails because 10 is not greater than 11.

Table B Check:

• For (-2, 23): Is \(\mathrm{23 \gt -4(-2) + 15}\)?
  \(\mathrm{23 \gt 8 + 15}\)
  \(\mathrm{23 \gt 23}\)? ✗ False

Table B fails immediately because 23 is not strictly greater than 23.

Table C Check:

• For (-2, 25): Is \(\mathrm{25 \gt -4(-2) + 15}\)?
  \(\mathrm{25 \gt 23}\)? ✓ True
• For (0, 16): Is \(\mathrm{16 \gt -4(0) + 15}\)?
  \(\mathrm{16 \gt 15}\)? ✓ True
• For (1, 12): Is \(\mathrm{12 \gt -4(1) + 15}\)?
  \(\mathrm{12 \gt 11}\)? ✓ True

All three pairs work for Table C!

4. CONSIDER ALL CASES by checking Table D for completeness

• For (-2, 24): Is \(\mathrm{24 \gt -4(-2) + 15}\)?
  \(\mathrm{24 \gt 23}\)? ✓ True
• For (0, 10): Is \(\mathrm{10 \gt -4(0) + 15}\)?
  \(\mathrm{10 \gt 15}\)? ✗ False

Table D fails because 10 is not greater than 15.

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often check only the first one or two pairs in each table instead of testing every single pair. For example, they might test Table A's first pair (-2, 25), see that \(\mathrm{25 \gt 23}\) is true, and mistakenly conclude Table A is correct without checking the remaining pairs. This leads them to select Choice A (Table A) even though the third pair (1, 10) actually fails the inequality test.

Second Most Common Error:

Conceptual confusion about strict inequality: Students may not understand that ">" means strictly greater than, not greater than or equal to. When they encounter Table B's first pair where \(\mathrm{23 \gt 23}\), they might think this is acceptable since 23 = 23. This conceptual gap causes them to select Choice B (Table B) instead of recognizing that 23 is not strictly greater than itself.

The Bottom Line:

This problem requires methodical checking of every data point combined with precise understanding of inequality symbols. Students who rush through the checking process or misunderstand the strict nature of ">" will consistently select incorrect answers.