Solve the inequality X - 1 / X - 3 less than 1 and represent the solution on number line and intervals

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.

Learn how to solve the rational inequality (X - 1) / (X - 3) < 1 with a step-by-step solution. This guide covers key concepts in inequalities, rational expressions, and representing the solution on a number line and using interval notation. Suitable for high school students in Grades 9-12.

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