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Find the LCD of the following rational equation

Let's examine the problem from the image.

The task is to find the Least Common Denominator (LCD) of the given rational equations. Below is the transcription of the equations from the image:

1. \(\frac{x}{x+3} + \frac{5}{x^2 - 9} = \frac{-1}{x-3}\)
2. \(\frac{x + 2}{x - 1} + \frac{1}{x^2 + 4x + 3} = \frac{-1}{x + 3}\)
3. \(\frac{x}{x^2 - 1} - \frac{2}{x + 1} = \frac{-1}{x - 1}\)
4. \(\frac{x - 3}{x + 1} + \frac{7}{x^2 - x - 12} = \frac{-1}{x - 3}\)

### Steps to find the LCD:

1. **Factor the denominators** in each equation.
2. **Identify the distinct factors** present in all the denominators.
3. **Multiply the distinct factors** to form the LCD.

Let's go through each equation step by step.

#### 1. \(\frac{x}{x+3} + \frac{5}{x^2 - 9} = \frac{-1}{x-3}\)
- \(x+3\) is already factored.
- \(x^2 - 9\) factors to \((x - 3)(x + 3)\).
- \(x - 3\) is already factored.

The distinct factors are \(x + 3\) and \(x - 3\).

**LCD**: \((x - 3)(x + 3)\)

#### 2. \(\frac{x + 2}{x - 1} + \frac{1}{x^2 + 4x + 3} = \frac{-1}{x + 3}\)
- \(x - 1\) is already factored.
- \(x^2 + 4x + 3\) factors to \((x + 3)(x + 1)\).
- \(x + 3\) is already factored.

The distinct factors are \(x - 1\), \(x + 3\), and \(x + 1\).

**LCD**: \((x - 1)(x + 3)(x + 1)\)

#### 3. \(\frac{x}{x^2 - 1} - \frac{2}{x + 1} = \frac{-1}{x - 1}\)
- \(x^2 - 1\) factors to \((x - 1)(x + 1)\).
- \(x + 1\) is already factored.
- \(x - 1\) is already factored.

The distinct factors are \(x - 1\) and \(x + 1\).

**LCD**: \((x - 1)(x + 1)\)

#### 4. \(\frac{x - 3}{x + 1} + \frac{7}{x^2 - x - 12} = \frac{-1}{x - 3}\)
- \(x + 1\) is already factored.
- \(x^2 - x - 12\) factors to \((x - 4)(x + 3)\).
- \(x - 3\) is already factored.

The distinct factors are \(x + 1\), \(x - 4\), and \(x + 3\).

**LCD**: \((x + 1)(x - 4)(x + 3)\)

### Summary of the LCDs for Each Equation:
1. \((x - 3)(x + 3)\)
2. \((x - 1)(x + 3)(x + 1)\)
3. \((x - 1)(x + 1)\)
4. \((x + 1)(x - 4)(x + 3)\)

Would you like to go over the steps in more detail, or do you have any questions?

Here are five related questions:
1. How do you factor a quadratic expression like \(x^2 - 9\)?
2. What is the importance of finding the LCD in rational equations?
3. How would you solve a rational equation after finding the LCD?
4. Can the LCD of two rational expressions be one of the original denominators?
5. What happens if you multiply both sides of a rational equation by the LCD?

**Tip:** When working with rational equations, always factor the denominators completely before finding the LCD. This ensures you include all necessary factors.

Learn how to find the Least Common Denominator (LCD) of rational equations step-by-step. This detailed guide covers multiple examples and explains the importance of factoring denominators. Suitable for high school students and those preparing for algebra exams.

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