The point \((5,3)\) in the xy-plane lies in a region bounded by the lines y = x and y =...

1. TRANSLATE the problem information

• Given information:
  • Point \((5,3)\) lies in a region
  • Region is bounded by lines \(\mathrm{y = x}\) and \(\mathrm{y = -x + 4}\)
  • Need to find which system of inequalities describes this region

• What this tells us: We need to test which side of each boundary line our point falls on.

2. INFER the approach

• To find which region contains a point, substitute the point's coordinates into each boundary line equation
• Check whether the point satisfies \(\mathrm{y \gt}\) (line equation) or \(\mathrm{y \lt}\) (line equation) for each boundary
• The correct answer will match both inequality conditions

3. SIMPLIFY by testing the first boundary line

• For \(\mathrm{y = x}\), substitute \((5,3)\):
• Is \(\mathrm{3 \gt 5}\) or \(\mathrm{3 \lt 5}\)?
• Since \(\mathrm{3 \lt 5}\), we have \(\mathrm{y \lt x}\)

4. SIMPLIFY by testing the second boundary line

• For \(\mathrm{y = -x + 4}\), substitute \((5,3)\):
• Calculate: \(\mathrm{-x + 4 = -5 + 4 = -1}\)
• Is \(\mathrm{3 \gt -1}\) or \(\mathrm{3 \lt -1}\)?
• Since \(\mathrm{3 \gt -1}\), we have \(\mathrm{y \gt -x + 4}\)

5. INFER the final answer

• Point \((5,3)\) satisfies: \(\mathrm{y \lt x}\) AND \(\mathrm{y \gt -x + 4}\)
• Looking at choices, only (C) matches both conditions

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the geometric language with the algebraic task. Instead of testing the given point in the boundary equations, they might try to solve the system of equations \(\mathrm{y = x}\) and \(\mathrm{y = -x + 4}\) to find intersection points, or attempt to graph the lines without using the specific point.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating \(\mathrm{-x + 4 = -5 + 4}\), potentially getting +1 instead of -1, or they reverse inequality directions when comparing 3 to their calculated value.

This may lead them to select Choice (D) \(\mathrm{(y \lt x}\) and \(\mathrm{y \lt -x + 4)}\) if they incorrectly conclude that \(\mathrm{3 \lt -x + 4}\).

The Bottom Line:

This problem tests whether students can connect the geometric concept of "a point lying in a bounded region" to the algebraic process of testing inequalities. Success requires methodical substitution and careful arithmetic with negative numbers.