For which of the following tables are all the values of x and their corresponding values of y solutions to...

1. TRANSLATE the problem requirements

• Given system of inequalities:
  • \(\mathrm{y \lt x}\) (y-values must be less than corresponding x-values)
  • \(\mathrm{x \lt 22}\) (x-values must be less than 22)
• Task: Find the table where ALL coordinate pairs satisfy BOTH conditions

2. APPLY CONSTRAINTS to understand what "solution" means

• For a table to be correct, every single (x,y) pair must satisfy both inequalities
• If even one pair fails either condition, that entire table is incorrect
• Both conditions must work together - this is a system

3. CONSIDER ALL CASES by checking each table systematically

Option A: Check every pair

• (19, 18): Is \(\mathrm{18 \lt 19}\)? Yes ✓ Is \(\mathrm{19 \lt 22}\)? Yes ✓
• (20, 19): Is \(\mathrm{19 \lt 20}\)? Yes ✓ Is \(\mathrm{20 \lt 22}\)? Yes ✓
• (21, 20): Is \(\mathrm{20 \lt 21}\)? Yes ✓ Is \(\mathrm{21 \lt 22}\)? Yes ✓

All pairs work!

Option B: Check every pair

• (19, 20): Is \(\mathrm{20 \lt 19}\)? No ❌

Stop here - this table fails immediately.

Option C: Check every pair

• (23, 22): Is \(\mathrm{22 \lt 23}\)? Yes ✓ Is \(\mathrm{23 \lt 22}\)? No ❌

Stop here - this table fails the \(\mathrm{x \lt 22}\) condition.

Option D: Check every pair

• (23, 24): Is \(\mathrm{24 \lt 23}\)? No ❌ Is \(\mathrm{23 \lt 22}\)? No ❌

Stop here - this table fails both conditions.

4. APPLY CONSTRAINTS to select the final answer

• Only Option A has all pairs satisfying both inequalities simultaneously

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students check that some pairs in a table work, but don't verify that ALL pairs work before selecting an answer.

For example, they might see that (19, 20) in Option B satisfies \(\mathrm{x \lt 22}\), notice that \(\mathrm{19 \lt 22}\) is true, and hastily select Option B without checking whether \(\mathrm{20 \lt 19}\). This leads them to select Choice B even though it fails the \(\mathrm{y \lt x}\) condition for every pair.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students might correctly check the first pair in each table but stop there instead of verifying every single pair.

They see (19, 18) works in Option A, (19, 20) fails in Option B, (23, 22) fails \(\mathrm{x \lt 22}\) in Option C, and (23, 24) fails both in Option D, then correctly choose A. However, if they had incomplete checking habits, they might miss other patterns and guess incorrectly.

The Bottom Line:

This problem tests systematic verification skills. Students must resist the urge to stop after finding one working example and instead methodically check every single data point against every single constraint.