Question:y gt x^2 - 6x + 10For which of the following tables are all the values of x and their...

1. TRANSLATE the problem requirements

• Given: The inequality \(\mathrm{y \gt x^2 - 6x + 10}\)
• Need to find: Which table has ALL coordinate pairs satisfying this inequality
• What this means: For each \(\mathrm{(x,y)}\) pair, the y-value must be greater than the result of \(\mathrm{x^2 - 6x + 10}\)

2. SIMPLIFY to find boundary values

Calculate \(\mathrm{x^2 - 6x + 10}\) for each x-value that appears in the tables:

• For \(\mathrm{x = 1}\):
  \(\mathrm{1^2 - 6(1) + 10}\)
  \(\mathrm{= 1 - 6 + 10}\)
  \(\mathrm{= 5}\)
• For \(\mathrm{x = 3}\):
  \(\mathrm{3^2 - 6(3) + 10}\)
  \(\mathrm{= 9 - 18 + 10}\)
  \(\mathrm{= 1}\)
• For \(\mathrm{x = 5}\):
  \(\mathrm{5^2 - 6(5) + 10}\)
  \(\mathrm{= 25 - 30 + 10}\)
  \(\mathrm{= 5}\)

This means we need: \(\mathrm{y \gt 5}\) when \(\mathrm{x = 1}\), \(\mathrm{y \gt 1}\) when \(\mathrm{x = 3}\), and \(\mathrm{y \gt 5}\) when \(\mathrm{x = 5}\)

3. APPLY CONSTRAINTS to check each table

Test every coordinate pair against the strict inequality:

Option A: \(\mathrm{(1,6): 6 \gt 5}\) ✓, \(\mathrm{(3,2): 2 \gt 1}\) ✓, \(\mathrm{(5,5): 5 \gt 5}\) ✗
Option B: \(\mathrm{(1,5): 5 \gt 5}\) ✗, \(\mathrm{(3,1): 1 \gt 1}\) ✗, \(\mathrm{(5,5): 5 \gt 5}\) ✗
Option C: \(\mathrm{(1,6): 6 \gt 5}\) ✓, \(\mathrm{(3,1): 1 \gt 1}\) ✗, \(\mathrm{(5,7): 7 \gt 5}\) ✓
Option D: \(\mathrm{(1,7): 7 \gt 5}\) ✓, \(\mathrm{(3,2): 2 \gt 1}\) ✓, \(\mathrm{(5,8): 8 \gt 5}\) ✓

Only Option D passes all three checks.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students confuse the strict inequality \(\mathrm{(\gt)}\) with greater than or equal to \(\mathrm{(\geq)}\), accepting coordinate pairs where y equals the boundary value.

For example, they might think \(\mathrm{(5,5)}\) satisfies \(\mathrm{y \gt 5}\) because "5 is close enough" or they misread the symbol. This leads them to incorrectly accept Option A, since it only fails on the technicality that 5 is not greater than 5.

This may lead them to select Choice A.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students make arithmetic errors when evaluating \(\mathrm{x^2 - 6x + 10}\), especially with the negative coefficient and order of operations.

For instance, they might calculate \(\mathrm{x = 3}\) as \(\mathrm{3^2 - 6(3) + 10 = 9 - 18 + 10 = 19}\) instead of 1, or forget to distribute the negative sign properly. This changes all their boundary comparisons and leads to random answer selection.

This causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can systematically check multiple conditions while maintaining precision with strict inequalities - a skill that requires both computational accuracy and careful attention to mathematical symbols.