Question:A quantity s is no less than 3 more than −4 times t.The value of t is −5.What is the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{s}\) is no less than 3 more than −4 times \(\mathrm{t}\)
  • \(\mathrm{t = -5}\)

• What this tells us: We need to convert "no less than" into mathematical notation. "No less than" means \(\geq\) (greater than or equal to).

2. TRANSLATE the complete inequality

• "3 more than −4 times t" becomes: \(\mathrm{-4t + 3}\)
• "s is no less than [this expression]" becomes: \(\mathrm{s \geq -4t + 3}\)

3. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{t = -5}\):
  \(\mathrm{s \geq -4(-5) + 3}\)
• Calculate:
  \(\mathrm{s \geq 20 + 3 = 23}\)

4. INFER the final answer

• Since \(\mathrm{s \geq 23}\), the least possible value of \(\mathrm{s}\) is exactly 23
• This is because \(\mathrm{s}\) can equal 23 or be any value greater than 23

Answer: 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE skill: Misinterpreting "no less than" as "less than"

Students often confuse the direction of inequality symbols, translating "no less than" as \(\lt\) instead of \(\geq\). This leads them to write \(\mathrm{s \lt -4t + 3}\), which after substitution gives \(\mathrm{s \lt 23}\). They might then incorrectly conclude there is no least value or pick an arbitrary number less than 23.

This leads to confusion and abandoning the systematic solution.

Second Most Common Error:

Weak SIMPLIFY execution: Making sign errors in arithmetic

Students correctly set up \(\mathrm{s \geq -4t + 3}\) but make calculation errors when substituting \(\mathrm{t = -5}\). Common mistakes include:

• Computing \(\mathrm{-4(-5)}\) as −20 instead of +20
• Adding \(\mathrm{20 + 3}\) incorrectly

This leads to getting \(\mathrm{s \geq 17}\) or another incorrect threshold, giving a wrong final answer.

The Bottom Line:

The key challenge is accurately translating English phrases involving inequalities into mathematical notation, then executing the arithmetic carefully with negative numbers.