Question:y gt |x - 3|For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the inequality into a checking process

• Given inequality: \(\mathrm{y \gt |x - 3|}\)
• Task: Find the table where ALL coordinate pairs (x, y) satisfy this inequality
• This means: For every row, the y-value must be strictly greater than \(\mathrm{|x - 3|}\)

2. SIMPLIFY by checking each table systematically

I'll evaluate \(\mathrm{|x - 3|}\) for each x-value and compare with the corresponding y-value.

Checking Table A:

• When x = 1: \(\mathrm{|1 - 3| = |-2| = 2}\), and y = 3. Since \(\mathrm{3 \gt 2}\) ✓
• When x = 3: \(\mathrm{|3 - 3| = |0| = 0}\), and y = 1. Since \(\mathrm{1 \gt 0}\) ✓
• When x = 5: \(\mathrm{|5 - 3| = |2| = 2}\), and y = 3. Since \(\mathrm{3 \gt 2}\) ✓

All three pairs work!

Checking Table B:

• When x = 1: \(\mathrm{|1 - 3| = 2}\), and y = 2. Since \(\mathrm{2 \gt 2}\)? No! \(\mathrm{2 = 2}\) ✗

Table B fails immediately.

Checking Table C:

• When x = 1: \(\mathrm{|1 - 3| = 2}\), and y = 3. Since \(\mathrm{3 \gt 2}\) ✓
• When x = 3: \(\mathrm{|3 - 3| = 0}\), and y = 0. Since \(\mathrm{0 \gt 0}\)? No! \(\mathrm{0 = 0}\) ✗

Table C fails at the second pair.

Checking Table D:

• When x = 1: \(\mathrm{|1 - 3| = 2}\), and y = 4. Since \(\mathrm{4 \gt 2}\) ✓
• When x = 3: \(\mathrm{|3 - 3| = 0}\), and y = 2. Since \(\mathrm{2 \gt 0}\) ✓
• When x = 5: \(\mathrm{|5 - 3| = 2}\), and y = 1. Since \(\mathrm{1 \gt 2}\)? No! \(\mathrm{1 \lt 2}\) ✗

Table D fails at the third pair.

3. APPLY CONSTRAINTS to select the final answer

Only Table A satisfies the strict inequality for all coordinate pairs.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students confuse > with ≥, thinking that \(\mathrm{y = |x - 3|}\) satisfies the inequality.

For example, in Table B when x = 1, they see \(\mathrm{|1 - 3| = 2}\) and y = 2, and incorrectly think "2 equals 2, so that works." They miss that the inequality requires y to be strictly greater than \(\mathrm{|x - 3|}\), not equal to it.

This may lead them to select Choice B or get confused when multiple tables seem to "work."

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when computing absolute values, especially with negative results inside the absolute value bars.

For instance, they might calculate \(\mathrm{|1 - 3| = -2}\) instead of \(\mathrm{|-2| = 2}\), leading to incorrect comparisons throughout their checking process.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand that inequalities require precise boundary conditions - "greater than" means the boundary itself doesn't count. The absolute value computation is straightforward, but the strict inequality comparison is where many students stumble.