During a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles...
GMAT Algebra : (Alg) Questions
During a portion of a flight, a small airplane's cruising speed varied between 150 miles per hour and 170 miles per hour. Which inequality best represents this situation, where \(\mathrm{s}\) is the cruising speed, in miles per hour, during this portion of the flight?
1. TRANSLATE the problem information
- Given information:
- Airplane's cruising speed varied between 150 mph and 170 mph
- \(\mathrm{s}\) represents the cruising speed in mph
- What "varied between" tells us: The speed had both a minimum and maximum value
2. INFER what "between" means mathematically
- In real-world contexts, "between" typically means inclusive of both endpoints
- This creates two constraints:
- Speed was at least 150 mph: \(\mathrm{s \geq 150}\)
- Speed was at most 170 mph: \(\mathrm{s \leq 170}\)
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"
Answer: D. \(\mathrm{150 \leq s \leq 170}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "between" or focus on only one constraint instead of recognizing both bounds.
Some students might think "between 150 and 170" means the difference is 20, leading them to consider Choice A (\(\mathrm{s \leq 20}\)). Others might focus only on the upper bound and select Choice C (\(\mathrm{s \leq 170}\)), missing that there's also a lower bound constraint.
This may lead them to select Choice A (\(\mathrm{s \leq 20}\)) or Choice C (\(\mathrm{s \leq 170}\))
Second Most Common Error:
Poor INFER reasoning about inequality direction: Students correctly identify that bounds exist but get confused about which direction the inequality symbols should face.
They might write something like \(\mathrm{s \leq 150 \leq 170}\), not recognizing that s must be greater than or equal to the lower bound. This confusion about inequality direction causes them to get stuck and guess.
The Bottom Line:
The key challenge is recognizing that "varied between" creates a range with both minimum and maximum constraints, requiring a compound inequality rather than a single inequality statement.