https://file.aiquickdraw.com/mathbot/file/1725841174114n3cvwmw1.jpg

Let's break down and solve the three equations from the image:

1. **First equation:**
   \[
   \frac{x + 3}{x + 5} - \frac{2}{x + 5} = 0
   \]
   We can combine the fractions since they have the same denominator:
   \[
   \frac{x + 3 - 2}{x + 5} = 0
   \]
   Simplifying:
   \[
   \frac{x + 1}{x + 5} = 0
   \]
   For a fraction to be zero, the numerator must be zero:
   \[
   x + 1 = 0
   \]
   Solving for \(x\):
   \[
   x = -1
   \]
   However, \(x\) cannot be \( -5 \) because it would make the denominator zero (undefined). So the solution is \(x = -1\).

2. **Second equation:**
   \[
   \frac{x}{4} + \frac{3}{x} = \frac{x + 6}{4}
   \]
   Multiply through by 4 to eliminate the fractions:
   \[
   x + \frac{12}{x} = x + 6
   \]
   Subtract \(x\) from both sides:
   \[
   \frac{12}{x} = 6
   \]
   Multiply both sides by \(x\):
   \[
   12 = 6x
   \]
   Solving for \(x\):
   \[
   x = 2
   \]

3. **Third equation:**
   \[
   \frac{x - 4}{x + 5} = \frac{36}{(x - 4)(x + 5)}
   \]
   Multiply both sides by \((x - 4)(x + 5)\) to eliminate the denominators:
   \[
   (x - 4)(x + 5) \times \frac{x - 4}{x + 5} = 36
   \]
   The terms \((x + 5)\) cancel on the left side, so:
   \[
   (x - 4)^2 = 36
   \]
   Taking the square root of both sides:
   \[
   x - 4 = \pm 6
   \]
   Solving for \(x\):
   \[
   x = 4 + 6 = 10 \quad \text{or} \quad x = 4 - 6 = -2
   \]
   So the solutions are \(x = 10\) or \(x = -2\).

---

Let me know if you want more detailed steps for any of the solutions or have questions.

### 5 Related Questions:
1. How would you handle an equation where the denominator has multiple terms and is zero?
2. How can you verify that a solution to a rational equation doesn't make the denominator zero?
3. What happens if both the numerator and denominator of a fraction are zero in an equation?
4. Can square roots have negative solutions, and how do we handle them in equations?
5. How can multiplying both sides by a term impact the solution set in rational equations?

### Tip:
Always check if any of the solutions make a denominator zero when dealing with rational equations to avoid extraneous solutions.

Solve for the values of x: 1) (x + 3)/(x + 5) - 2/(x + 5) = 0, 2) x/4 + 3/x = (x + 6)/4, 3) (x - 4)/(x + 5) = 36/((x - 4)(x + 5))

Learn how to solve rational and quadratic equations, including (x + 3)/(x + 5) - 2/(x + 5) = 0, x/4 + 3/x = (x + 6)/4, and (x - 4)/(x + 5) = 36/((x - 4)(x + 5)). Follow the detailed solutions and understand key algebraic concepts for high school students.

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