In the xy-plane, the point \(\mathrm{(t, -12)}\) is a solution to the system of inequalities shown below.3t + y gt...

1. TRANSLATE the problem information

• Given information:
  • Point \(\mathrm{(t, -12)}\) is a solution to both inequalities
  • This means when we substitute \(\mathrm{y = -12}\), both inequalities must be true
• What this tells us: We need to substitute \(\mathrm{y = -12}\) into both inequalities and solve for t

2. SIMPLIFY the first inequality

• Starting with: \(\mathrm{3t + y \gt -20}\)
• Substitute \(\mathrm{y = -12}\): \(\mathrm{3t + (-12) \gt -20}\)
• Simplify: \(\mathrm{3t - 12 \gt -20}\)
• Add 12 to both sides: \(\mathrm{3t \gt -8}\)
• Divide by 3: \(\mathrm{t \gt -8/3}\)
• Convert to decimal (use calculator): \(\mathrm{-8/3 = -2.67}\)
• So: \(\mathrm{t \gt -2.67}\)

3. SIMPLIFY the second inequality

• Starting with: \(\mathrm{2t - y \lt 10}\)
• Substitute \(\mathrm{y = -12}\): \(\mathrm{2t - (-12) \lt 10}\)
• Careful with signs: \(\mathrm{2t + 12 \lt 10}\)
• Subtract 12 from both sides: \(\mathrm{2t \lt -2}\)
• Divide by 2: \(\mathrm{t \lt -1}\)

4. INFER the combined constraint

• From step 2: \(\mathrm{t \gt -2.67}\)
• From step 3: \(\mathrm{t \lt -1}\)
• Combined: \(\mathrm{-2.67 \lt t \lt -1}\)

5. APPLY CONSTRAINTS to check answer choices

• (A) −5: Is \(\mathrm{-5 \gt -2.67}\)? No, so this fails
• (B) −3: Is \(\mathrm{-3 \gt -2.67}\)? No, so this fails
• (C) −2: Is \(\mathrm{-2 \gt -2.67}\)? Yes. Is \(\mathrm{-2 \lt -1}\)? Yes. This works!
• (D) 0: Is \(\mathrm{0 \lt -1}\)? No, so this fails

Answer: C (−2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Sign errors when handling the second inequality, especially with \(\mathrm{2t - (-12)}\). Students often write \(\mathrm{2t - 12 \lt 10}\) instead of \(\mathrm{2t + 12 \lt 10}\), leading to \(\mathrm{t \lt 11}\) instead of \(\mathrm{t \lt -1}\).

With \(\mathrm{t \gt -2.67}\) and the incorrect \(\mathrm{t \lt 11}\), students would think all answer choices work and get confused, often guessing or selecting the first choice that "looks reasonable."

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students solve both inequalities correctly but fail to check both conditions simultaneously. They might only verify that a choice satisfies one inequality, missing that it must satisfy both.

For example, they might see that −5 satisfies \(\mathrm{t \lt -1}\) from the second inequality and select (A) −5 without checking the first constraint \(\mathrm{t \gt -2.67}\).

The Bottom Line:

This problem tests whether students can systematically work through a multi-step constraint problem. Success requires careful algebraic manipulation and methodical checking of compound inequalities.