Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.)
Subtracting $2$ from both sides, we get \[\frac{x-1}{x-3} - 2 \ge 0,\]or \[\frac{x-1 - 2(x-3)}{x-3} = \frac{-x+5}{x-3} \ge 0.\]Negating both sides, we have \[\frac{x-5}{x-3} \le 0.\]Letting $f(x) = \frac{x-5}{x-3},$ we make a sign table with the two factors $x-5$ and $x-3$: \begin{tabular}{c|cc|c} &$x-5$ &$x-3$ &$f(x)$ \\ \hline$x<3$ &$-$&$-$&$+$\\ [.1cm]$3<x<5$ &$-$&$+$&$-$\\ [.1cm]$x>5$ &$+$&$+$&$+$\\ [.1cm]\end{tabular}We see that $f(x) \le 0$ when $3 < x < 5,$ as well as at the endpoint $x=5.$ Therefore, the solution set is the interval $\boxed{ (3, 5] }.$
(3,5]
Level 3
Intermediate Algebra
Problem: Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.)
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the inequality \(\frac{x-1}{x-3} \ge 2\), first, we need to manipulate the inequality to isolate \(x\). Start by subtracting 2 from both sides:

\[
\frac{x-1}{x-3} - 2 \ge 0.
\]

Next, express 2 with a common denominator:

\[
\frac{x-1}{x-3} - \frac{2(x-3)}{x-3} \ge 0.
\]

This simplifies to:

\[
\frac{x-1 - 2(x-3)}{x-3} \ge 0.
\]

Distributing the -2 gives:

\[
\frac{x-1 - 2x + 6}{x-3} \ge 0.
\]

Combining like terms results in:

\[
\frac{-x + 5}{x-3} \ge 0.
\]

This can be rewritten as:

\[
\frac{5 - x}{x-3} \ge 0.
\]

Next, identify the critical points by setting the numerator and denominator to zero:

1. \(5 - x = 0 \Rightarrow x = 5\)
2. \(x - 3 = 0 \Rightarrow x = 3\)

These critical points divide the number line into intervals: \((-\infty, 3)\), \((3, 5)\), and \((5, \infty)\). We will test each interval to determine where the inequality holds.

1. For the interval \((-\infty, 3)\), choose \(x = 0\):
   \[
   \frac{5 - 0}{0 - 3} = \frac{5}{-3} < 0.
   \]
   This interval does not satisfy the inequality.

2. For the interval \((3, 5)\), choose \(x = 4\):
   \[
   \frac{5 - 4}{4 - 3} = \frac{1}{1} > 0.
   \]
   This interval satisfies the inequality.

3. For the interval \((5, \infty)\), choose \(x = 6\):
   \[
   \frac{5 - 6}{6 - 3} = \frac{-1}{3} < 0.
   \]
   This interval does not satisfy the inequality.

Next, check the endpoints. At \(x = 3\), the expression is undefined, so it cannot be included. At \(x = 5\):

\[
\frac{5 - 5}{5 - 3} = \frac{0}{2} = 0,
\]
which satisfies the inequality.

Thus, the solution set is the interval \((3, 5]\). The final answer is \((3, 5]\).<|END_OF_LATENT|> The final answer is $(3,5]$.