Discussion
If x satisfies the inequality \(\frac{x - 3}{x - 5} \gt 3\) then \(x\) lies in the interval
Answer: C
\(x\) lies in \((5, 6)\).
Do not multiply across by \(x - 5\), whose sign is unknown. Bring everything to one side instead:
\[ \begin{aligned} \frac{x - 3}{x - 5} - 3 &\gt 0 \\ \frac{x - 3 - 3x + 15}{x - 5} &\gt 0 \\ \frac{2(6 - x)}{x - 5} &\gt 0 \end{aligned} \]
The fraction is positive when the numerator and denominator have the same sign: both positive needs \(x \lt 6\) and \(x \gt 5\); both negative needs \(x \gt 6\) and \(x \lt 5\), impossible. So \(5 \lt x \lt 6\).
Check \(x = 5.5\): \(2.5/0.5 = 5 \gt 3\). Options (3,8) and (5,8) include points such as 7, where the fraction is \(4/2 = 2\), not greater than 3.
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