An ordered pair \(\mathrm{(x, y)}\) is defined by x = t - 1 and y = 2t + 1, where...

1. TRANSLATE the problem information

• Given information:
  • Parametric equations: \(\mathrm{x = t - 1}\) and \(\mathrm{y = 2t + 1}\)
  • System constraints: \(\mathrm{y \geq -2x + 1}\) and \(\mathrm{3x - y \lt 5}\)
  • Need to find which t value satisfies both inequalities
• What this tells us: We need to substitute the parametric expressions into both inequalities to find the valid range for t.

2. TRANSLATE the first inequality

• Substitute \(\mathrm{x = t - 1}\) and \(\mathrm{y = 2t + 1}\) into \(\mathrm{y \geq -2x + 1}\):
  \(\mathrm{2t + 1 \geq -2(t - 1) + 1}\)

3. SIMPLIFY the first inequality

• Expand the right side:
  \(\mathrm{2t + 1 \geq -2t + 2 + 1}\)
  \(\mathrm{2t + 1 \geq -2t + 3}\)
• Collect like terms:
  \(\mathrm{2t + 2t \geq 3 - 1}\)
  \(\mathrm{4t \geq 2}\)
  \(\mathrm{t \geq 0.5}\)

4. TRANSLATE the second inequality

• Substitute \(\mathrm{x = t - 1}\) and \(\mathrm{y = 2t + 1}\) into \(\mathrm{3x - y \lt 5}\):
  \(\mathrm{3(t - 1) - (2t + 1) \lt 5}\)

5. SIMPLIFY the second inequality

• Expand:
  \(\mathrm{3t - 3 - 2t - 1 \lt 5}\)
  \(\mathrm{t - 4 \lt 5}\)
  \(\mathrm{t \lt 9}\)

6. APPLY CONSTRAINTS to find valid t range

• Both inequalities must be satisfied simultaneously:
  • From inequality 1: \(\mathrm{t \geq 0.5}\)
  • From inequality 2: \(\mathrm{t \lt 9}\)
  • Combined: \(\mathrm{0.5 \leq t \lt 9}\)

7. APPLY CONSTRAINTS to check answer choices

• (A) \(\mathrm{t = -2}\): No, since \(\mathrm{-2 \lt 0.5}\)
• (B) \(\mathrm{t = 0}\): No, since \(\mathrm{0 \lt 0.5}\)
• (C) \(\mathrm{t = 1}\): Yes, since \(\mathrm{0.5 \leq 1 \lt 9}\) ✓
• (D) \(\mathrm{t = 9}\): No, since we need \(\mathrm{t \lt 9}\) (strict inequality)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often substitute incorrectly or make sign errors when dealing with expressions like \(\mathrm{-2(t - 1)}\).

A common mistake is writing \(\mathrm{-2(t - 1) = -2t - 2}\) instead of \(\mathrm{-2t + 2}\). This leads to the wrong inequality \(\mathrm{2t + 1 \geq -2t - 2 + 1}\), which gives \(\mathrm{2t + 1 \geq -2t - 1}\), so \(\mathrm{4t \geq -2}\), yielding \(\mathrm{t \geq -0.5}\). Combined with \(\mathrm{t \lt 9}\), this would make all answer choices except (A) appear valid, causing confusion and likely guessing.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students solve each inequality correctly but fail to properly combine the conditions, especially mishandling the boundary conditions.

Some students might incorrectly include \(\mathrm{t = 9}\) as valid, not recognizing that the strict inequality \(\mathrm{3x - y \lt 5}\) means \(\mathrm{t \lt 9}\) (not \(\mathrm{t \leq 9}\)). This could lead them to select Choice D (9) as a valid answer.

The Bottom Line:

This problem tests whether students can systematically work with parametric representations in constraint systems. The key challenge is maintaining algebraic accuracy while translating between different mathematical representations and properly handling multiple simultaneous conditions.