A system of inequalities is given by y geq x + 1 and x leq 6. Which of the following...

1. TRANSLATE the problem requirements

• Given system: \(\mathrm{y \geq x + 1}\) AND \(\mathrm{x \leq 6}\)
• Need: A set where ALL ordered pairs satisfy BOTH inequalities
• This means every single point must pass both tests

2. APPLY CONSTRAINTS systematically to each choice

Choice A: {(4, 5), (6, 6), (6, 8)}

• Test \(\mathrm{(4, 5)}\): Does \(\mathrm{5 \geq 4 + 1}\)? Yes, \(\mathrm{5 \geq 5}\) ✓. Does \(\mathrm{4 \leq 6}\)? Yes ✓
• Test \(\mathrm{(6, 6)}\): Does \(\mathrm{6 \geq 6 + 1}\)? No, \(\mathrm{6 \ngeq 7}\) ✗

Since \(\mathrm{(6, 6)}\) fails the first inequality, Choice A is eliminated.

Choice B: {(4, 5), (6, 7), (7, 8)}

• Test \(\mathrm{(7, 8)}\): Does \(\mathrm{7 \leq 6}\)? No ✗

Since \(\mathrm{(7, 8)}\) fails the second inequality, Choice B is eliminated.

3. CONSIDER ALL CASES for remaining choices

Choice C: {(3, 4), (6, 8), (-1, 0)}

• Test \(\mathrm{(3, 4)}\): \(\mathrm{4 \geq 3 + 1}\) → \(\mathrm{4 \geq 4}\) ✓ and \(\mathrm{3 \leq 6}\) ✓
• Test \(\mathrm{(6, 8)}\): \(\mathrm{8 \geq 6 + 1}\) → \(\mathrm{8 \geq 7}\) ✓ and \(\mathrm{6 \leq 6}\) ✓
• Test \(\mathrm{(-1, 0)}\): \(\mathrm{0 \geq -1 + 1}\) → \(\mathrm{0 \geq 0}\) ✓ and \(\mathrm{-1 \leq 6}\) ✓

All three points satisfy both inequalities! But let's verify the other choices are truly eliminated.

Choice D: {(-2, -1), (1, 2), (6, 6)}

• We already know \(\mathrm{(6, 6)}\) fails: \(\mathrm{6 \ngeq 7}\), so Choice D is out.

Choice E: {(0, 0), (2, 3), (3, 4)}

• Test \(\mathrm{(0, 0)}\): Does \(\mathrm{0 \geq 0 + 1}\)? No, \(\mathrm{0 \ngeq 1}\) ✗

Choice E is eliminated.

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete APPLY CONSTRAINTS: Students check inequality conditions but don't verify ALL points in a set

Many students will check the first point or two in each choice, find ones that work, and immediately select that answer without completing the verification. For example, they might check \(\mathrm{(4, 5)}\) in Choice A, see it works, and select A without testing \(\mathrm{(6, 6)}\).

This leads them to select Choice A {(4, 5), (6, 6), (6, 8)} or Choice D {(-2, -1), (1, 2), (6, 6)} since both contain points that do satisfy the system.

Second Most Common Error:

Weak CONSIDER ALL CASES: Students only test one inequality per point instead of both

Some students might systematically go through each point but only check \(\mathrm{y \geq x + 1}\) OR only check \(\mathrm{x \leq 6}\), missing violations of the other condition. This incomplete checking can make invalid choices seem correct.

This may lead them to select Choice B {(4, 5), (6, 7), (7, 8)} since all points satisfy \(\mathrm{y \geq x + 1}\), but they miss that \(\mathrm{(7, 8)}\) violates \(\mathrm{x \leq 6}\).

The Bottom Line:

This problem tests methodical verification skills. Students must resist the urge to stop checking once they find points that work, and instead systematically verify that every single point in a set satisfies both conditions of the system.