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

1. TRANSLATE the inequality requirement

• Given: \(\mathrm{y < 6x + 2}\)
• We need ALL values in a table to satisfy this inequality
• This means for each x-value, the corresponding y-value must be less than \(\mathrm{6x + 2}\)

2. INFER the systematic approach

• Since all tables use the same x-values (3, 5, 7), calculate the boundary for each:
• For each x, find what \(\mathrm{6x + 2}\) equals, then check if the table's y is less than that value

3. SIMPLIFY to find the boundaries

• For \(\mathrm{x = 3}\): \(\mathrm{6(3) + 2 = 18 + 2 = 20}\), so \(\mathrm{y < 20}\)
• For \(\mathrm{x = 5}\): \(\mathrm{6(5) + 2 = 30 + 2 = 32}\), so \(\mathrm{y < 32}\)
• For \(\mathrm{x = 7}\): \(\mathrm{6(7) + 2 = 42 + 2 = 44}\), so \(\mathrm{y < 44}\)

4. APPLY CONSTRAINTS to each table

Check each table systematically:

Choice A: All y-values equal the boundaries (20, 32, 44) - not less than, so fails
Choice B: \(\mathrm{y = 36}\) when \(\mathrm{x = 5}\), but \(\mathrm{36 > 32}\), so fails
Choice C: All y-values (16, 28, 40) are less than their boundaries (20, 32, 44) ✓
Choice D: All y-values exceed their boundaries, so fails

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret '\(\mathrm{y < 6x + 2}\)' as '\(\mathrm{y \leq 6x + 2}\)' or don't understand that the inequality must be strict.

They see that Choice A has y-values of exactly 20, 32, and 44, which equal \(\mathrm{6x + 2}\) for each corresponding x, and incorrectly think this satisfies the inequality. Since these values equal (rather than are less than) the expression, Choice A fails the strict inequality test.

This may lead them to select Choice A (with y-values 20, 32, 44).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{6x + 2}\), getting wrong boundary values.

For example, calculating \(\mathrm{6(5) + 2}\) as 28 instead of 32, or \(\mathrm{6(7) + 2}\) as 40 instead of 44. With wrong boundaries, they can't properly evaluate which tables satisfy the inequality.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand strict inequality notation and can systematically check multiple conditions. The key insight is that 'less than' means the y-value cannot equal the boundary - it must be strictly smaller.