To solve the inequality:

\[
\frac{x - 1}{x - 3} > -1
\]

### Step 1: Isolate the rational expression
First, add \(1\) to both sides to get rid of the \(-1\) on the right-hand side:

\[
\frac{x - 1}{x - 3} + 1 > 0
\]

### Step 2: Simplify the expression
To simplify, combine the two terms on the left-hand side. Rewrite \(1\) as \(\frac{x - 3}{x - 3}\) to have a common denominator:

\[
\frac{x - 1}{x - 3} + \frac{x - 3}{x - 3} = \frac{(x - 1) + (x - 3)}{x - 3} = \frac{2x - 4}{x - 3}
\]

Thus, the inequality becomes:

\[
\frac{2x - 4}{x - 3} > 0
\]

### Step 3: Solve the inequality
Now, solve the inequality \(\frac{2x - 4}{x - 3} > 0\). This rational expression will change sign where the numerator and denominator are zero. The critical points are:

- Numerator: \(2x - 4 = 0 \Rightarrow x = 2\)
- Denominator: \(x - 3 = 0 \Rightarrow x = 3\)

### Step 4: Analyze the intervals
Now divide the number line into intervals based on the critical points \(x = 2\) and \(x = 3\). Test points in each interval to determine where the expression is positive.

- For \(x < 2\), choose \(x = 0\): \(\frac{2(0) - 4}{0 - 3} = \frac{-4}{-3} = \frac{4}{3} > 0\)
- For \(2 < x < 3\), choose \(x = 2.5\): \(\frac{2(2.5) - 4}{2.5 - 3} = \frac{5 - 4}{-0.5} = \frac{1}{-0.5} = -2 < 0\)
- For \(x > 3\), choose \(x = 4\): \(\frac{2(4) - 4}{4 - 3} = \frac{8 - 4}{1} = \frac{4}{1} = 4 > 0\)

### Step 5: Determine the solution
The expression \(\frac{2x - 4}{x - 3} > 0\) is positive for \(x < 2\) and \(x > 3\). At \(x = 2\) and \(x = 3\), the expression is zero or undefined.

### Final solution:
\[
x \in (-\infty, 2) \cup (3, \infty)
\]

This is the solution to the inequality.

Would you like further details or have any questions?

### Related questions:
1. How do you solve similar rational inequalities?
2. Can this inequality be represented graphically?
3. What are the steps for solving absolute value inequalities?
4. How does multiplying both sides of an inequality by a negative number affect the result?
5. What are the common methods for solving quadratic inequalities?

### Tip:
Always check the sign of both the numerator and denominator in different intervals to solve rational inequalities effectively.

Learn how to solve the rational inequality (x - 1)/(x - 3) > -1 step by step. This detailed explanation covers simplifying rational expressions, finding critical points, and performing sign analysis. Ideal for students in Grades 10-12.

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