\(\mathrm{y For which of the following tables are all the values of x and their corresponding values of y solutions...

1. TRANSLATE the problem requirement

• Given: The inequality \(\mathrm{y < 5x + 6}\)
• Need to find: Which table has ALL ordered pairs (x,y) satisfying this inequality
• This means: For each x value, the corresponding y must be less than \(\mathrm{5x + 6}\)

2. INFER the checking strategy

• We must test every single ordered pair in each table
• If even one pair fails the inequality, that entire table is wrong
• We can stop checking a table as soon as we find one failing pair

3. SIMPLIFY by checking each table systematically

Table A:

• When \(\mathrm{x = 3}\): Calculate \(\mathrm{5(3) + 6 = 21}\). Is \(\mathrm{17 < 21}\)? Yes ✓
• When \(\mathrm{x = 5}\): Calculate \(\mathrm{5(5) + 6 = 31}\). Is \(\mathrm{27 < 31}\)? Yes ✓
• When \(\mathrm{x = 7}\): Calculate \(\mathrm{5(7) + 6 = 41}\). Is \(\mathrm{37 < 41}\)? Yes ✓

All three pairs work!

Table B:

• When \(\mathrm{x = 3}\): Calculate \(\mathrm{5(3) + 6 = 21}\). Is \(\mathrm{17 < 21}\)? Yes ✓
• When \(\mathrm{x = 5}\): Calculate \(\mathrm{5(5) + 6 = 31}\). Is \(\mathrm{35 < 31}\)? No ✗

Stop here - this table fails.

Table C:

• When \(\mathrm{x = 3}\): Calculate \(\mathrm{5(3) + 6 = 21}\). Is \(\mathrm{25 < 21}\)? No ✗

Stop here - this table fails.

Table D:

• When \(\mathrm{x = 3}\): Calculate \(\mathrm{5(3) + 6 = 21}\). Is \(\mathrm{21 < 21}\)? No ✗

Stop here - this table fails.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak understanding of strict inequality (<): Students confuse "less than" with "less than or equal to," thinking that \(\mathrm{y = 21}\) satisfies \(\mathrm{y < 21}\). When they check Table D and see that when \(\mathrm{x = 3}\), both y and \(\mathrm{5x + 6}\) equal 21, they incorrectly conclude this satisfies the inequality.

This may lead them to select Choice D (21, 31, 41) or causes confusion about which tables are valid.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic mistakes when calculating \(\mathrm{5x + 6}\), particularly with \(\mathrm{5(7) + 6}\). Getting 42 instead of 41, or 30 instead of 31, leads to wrong comparisons and incorrect elimination of valid tables.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires careful, systematic checking of every single ordered pair. Students who try to take shortcuts or who aren't precise about the strict inequality symbol will miss the correct answer.