Question

If \(x \neq 11\) satisfies the inequality \(\frac{2x - 21}{x - 11} \geq 3\), then \(x\) lies in the interval

• A. \((-\infty, 11)\)
• B. \((11, 12]\)
• C. \((11, 12)\)
• D. \((11, \infty)\)
• E. \([12, \infty)\)

Hint:

Always consider critical points where numerator or denominator equals zero.

Answer 1

The Correct Option is B

Step 1: Concept:
• Bring all terms to one side and take a common denominator.
Step 2: Simplification:
• Given inequality: \(\frac{2x - 21}{x - 11} - 3 \geq 0\)
• Combine into a single fraction: \(\frac{2x - 21 - 3(x - 11)}{x - 11} \geq 0\)
• Simplify numerator: \(2x - 21 - 3x + 33 = -x + 12\)
• So, inequality becomes: \(\frac{12 - x}{x - 11} \geq 0\)
Step 3: Critical Points & Sign Analysis:
• Critical points: \(x = 11\) (denominator), \(x = 12\) (numerator)
• Check intervals:
• \(x < 11\): numerator \(> 0\), denominator \(< 0\) \(\Rightarrow\) expression \(< 0\)
• \(11 < x < 12\): numerator \(> 0\), denominator \(> 0\) \(\Rightarrow\) expression \(> 0\)
• \(x > 12\): numerator \(< 0\), denominator \(> 0\) \(\Rightarrow\) expression \(< 0\)
• At \(x = 12\), value = 0 (included)
• At \(x = 11\), undefined (excluded)
Step 4: Final Answer:
• Solution set: \(x \in (11, 12]\)