During a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles...

1. TRANSLATE the problem information

• Given information:
  • The airplane's cruising speed varied between 150 mph and 170 mph
  • \(\mathrm{s}\) represents the cruising speed in miles per hour
• What this tells us: We need to find an inequality that captures this range

2. INFER what "between" means mathematically

• "Between 150 and 170" establishes two separate constraints:
  • The speed was never less than 150 mph → \(\mathrm{s \geq 150}\)
  • The speed was never more than 170 mph → \(\mathrm{s \leq 170}\)
• These must both be true simultaneously

3. TRANSLATE into compound inequality notation

• Combine both constraints: \(\mathrm{150 \leq s \leq 170}\)
• This reads as "s is greater than or equal to 150 AND less than or equal to 170"

4. Verify against answer choices

• Choice D matches our compound inequality exactly

Answer: D. \(\mathrm{150 \leq s \leq 170}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students partially translate "between 150 and 170," capturing only one boundary instead of recognizing that "between" establishes both minimum AND maximum constraints.

They might think "between 150 and 170" only means "no more than 170" and select Choice C (\(\mathrm{s \leq 170}\)), missing that the speed also cannot go below 150.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what numerical relationship "between 150 and 170" represents, possibly focusing on the difference (\(\mathrm{170 - 150 = 20}\)) rather than the range itself.

This confusion about the meaning of "between" leads to uncertainty and guessing among the available choices.

The Bottom Line:

The key challenge is recognizing that "between" in mathematical contexts means establishing both a lower AND upper boundary, requiring a compound inequality rather than a simple one-sided constraint.