The given inequality is:

$\frac{X - 1}{X - 3} < 1$

To solve this inequality, we follow these steps:

Step 1: Bring everything to one side

We subtract $1$ from both sides:

$\frac{X - 1}{X - 3} - 1 < 0$

Step 2: Combine the terms under a common denominator

We express $1$ as $\frac{X - 3}{X - 3}$ and subtract the fractions:

$\frac{X - 1}{X - 3} - \frac{X - 3}{X - 3} = \frac{(X - 1) - (X - 3)}{X - 3} = \frac{X - 1 - X + 3}{X - 3} = \frac{2}{X - 3}$

Thus, the inequality becomes:

$\frac{2}{X - 3} < 0$

Step 3: Solve the inequality

The fraction $\frac{2}{X - 3}$ is negative when the denominator is negative. Therefore, we need:

$X - 3 < 0 \quad \text{or} \quad X < 3$

Step 4: Represent the solution

The solution to the inequality is $X < 3$.

Interval notation:

The solution is $(-\infty, 3)$.

Number line representation:

• Draw a number line.
• Mark $X = 3$ with an open circle because $X = 3$ is not included in the solution (the inequality is strict).
• Shade the region to the left of $3$, indicating $X < 3$.

Would you like the graph representation or further clarifications on this solution?

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Here are 5 related questions:

1. What happens if the inequality was $\frac{X - 1}{X - 3} > 1$?
2. How do you solve inequalities involving quadratic expressions?
3. Can you graph rational functions and their asymptotes?
4. What changes in the solution if the inequality were $\leq 1$?
5. How do you handle inequalities involving absolute values?

Tip: Always check for points where the denominator of a rational expression is zero when solving inequalities.