y gt 13x - 18For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the inequality into testable conditions

• Given: \(\mathrm{y \gt 13x - 18}\)
• We have four tables with the same x-values: 3, 5, and 8
• We need to find which table has ALL y-values satisfying the inequality

2. INFER the solution approach

• For each x-value, calculate what the minimum y-value should be
• Then check if each table's y-values exceed these minimums
• Remember: ALL pairs in the table must work, not just some

3. Calculate the y-thresholds for each x-value

For \(\mathrm{x = 3}\):

\(\mathrm{y \gt 13(3) - 18}\)

\(\mathrm{y \gt 39 - 18}\)

\(\mathrm{y \gt 21}\)

For \(\mathrm{x = 5}\):

\(\mathrm{y \gt 13(5) - 18}\)

\(\mathrm{y \gt 65 - 18}\)

\(\mathrm{y \gt 47}\)

For \(\mathrm{x = 8}\):

\(\mathrm{y \gt 13(8) - 18}\)

\(\mathrm{y \gt 104 - 18}\)

\(\mathrm{y \gt 86}\)

4. APPLY CONSTRAINTS to test each table systematically

Table A: (3,21), (5,47), (8,86)

• Is \(\mathrm{21 \gt 21}\)? No (equal, not greater)
• Table A fails immediately

Table B: (3,26), (5,42), (8,86)

• Is \(\mathrm{26 \gt 21}\)? Yes ✓
• Is \(\mathrm{42 \gt 47}\)? No (less than required)
• Table B fails

Table C: (3,16), (5,42), (8,81)

• Is \(\mathrm{16 \gt 21}\)? No (less than required)
• Table C fails immediately

Table D: (3,26), (5,52), (8,91)

• Is \(\mathrm{26 \gt 21}\)? Yes ✓
• Is \(\mathrm{52 \gt 47}\)? Yes ✓
• Is \(\mathrm{91 \gt 86}\)? Yes ✓
• All conditions satisfied!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS: Students treat \(\mathrm{\gt}\) as \(\mathrm{\geq}\) and accept boundary values as solutions

Students often think "21 is basically the same as 21" or "close enough counts." They incorrectly conclude that \(\mathrm{y = 21}\) satisfies \(\mathrm{y \gt 21}\), leading them to consider Table A as valid. When multiple tables seem to "work" under this flawed reasoning, they may guess or pick the first one they checked.

This may lead them to select Choice A (with values 21, 47, 86) because these are the exact boundary values that feel "correct."

Second Most Common Error:

Inadequate TRANSLATE execution: Students make arithmetic errors when calculating the y-thresholds

Students might incorrectly compute \(\mathrm{13(5) - 18 = 65 - 18 = 57}\) instead of 47, or make similar calculation mistakes. With wrong thresholds, they'll evaluate tables against incorrect criteria and reach the wrong conclusion.

This leads to confusion and random answer selection since their systematic checking produces contradictory results.

The Bottom Line:

This problem tests precision with inequalities - the difference between \(\mathrm{\gt}\) and \(\mathrm{\geq}\) is crucial, and even small calculation errors derail the entire solution. Students must be methodical in both their arithmetic and their understanding of strict inequalities.