Question

If \( 2 \) is a solution of the inequality \( \frac{x-a}{a-2x}<-3 \), then \( a \) must lie in the interval:

• A. \( (4,5) \)
• B. \( (2,5) \)
• C. \( (4,10) \)
• D. \( (2,10) \)
• E. \( (0,10) \)

Hint:

When solving inequalities involving fractions and a variable, remember to carefully consider the implications of multiplying or dividing by expressions that contain variables. Always check the sign of the expression being multiplied or divided to ensure the inequality's direction is correctly maintained. Additionally, consider the domain restrictions imposed by denominators to avoid undefined expressions.

Answer 1

The Correct Option is A

First, substitute \( x = 2 \) into the inequality and simplify: \[ \frac{2-a}{a-4}<-3. \] Multiply both sides by \( a-4 \) (assuming \( a \neq 4 \)) to avoid reversing the inequality: \[ 2 - a<-3(a - 4). \] Expanding and simplifying yields: \[ 2 - a<-3a + 12 \quad \Rightarrow \quad 2a<10 \quad \Rightarrow \quad a<5. \] 
Now, because the inequality assumes \( a - 4 \) is positive (so we do not reverse the inequality sign when multiplying), it implies \( a>4 \).
Conclusion: Combining \( a>4 \) and \( a<5 \), we find that \( a \) must lie in the interval \( (4,5) \), matching option (A).