https://file.aiquickdraw.com/mathbot/file/1725840552296b9y5asrn.jpg

Let's analyze each equation from the image to determine which are **false**.

### 1. \( \frac{4 + \sqrt{x^2 + 5}}{4} = 1 + \frac{\sqrt{x^2 + 5}}{4} \)

- This is a false equation. Simplifying the left-hand side:
  \[
  \frac{4 + \sqrt{x^2 + 5}}{4} = \frac{4}{4} + \frac{\sqrt{x^2 + 5}}{4} = 1 + \frac{\sqrt{x^2 + 5}}{4}
  \]
  So the two sides are indeed equal, **meaning it is not false**.

### 2. \( -( -6 )^2 = -36 \)

- This is a **false** statement. Here's why:
  \[
  ( -6 )^2 = 36, \quad \text{so} \quad -( -6 )^2 = -36
  \]
  Therefore, the given equation is true, which contradicts the idea that this is a false equation.

### 3. \( \frac{x + 6}{x + y} = \frac{6}{x + y} \)

- This is a **false** equation. The left-hand side cannot be simplified to the right-hand side unless \( x = 0 \), which is not universally true.

### 4. \( \frac{1}{x} + \frac{9}{x + 9} = \frac{x + 9}{x} \)

- This is a **false** equation. Let's verify:
  \[
  \frac{1}{x} + \frac{9}{x + 9} \neq \frac{x + 9}{x}
  \]
  There's no simplification that would make the two sides equal.

### Conclusion:
The false equations are:
- \( \frac{x + 6}{x + y} = \frac{6}{x + y} \)
- \( \frac{1}{x} + \frac{9}{x + 9} = \frac{x + 9}{x} \)

Would you like more details or have any questions?

### 5 Relative Questions:
1. How do you simplify fractions involving square roots?
2. Why does the placement of negative signs outside parentheses change results in exponents?
3. Can you explain the properties of proportions when simplifying algebraic fractions?
4. What are the steps to combine terms with different denominators?
5. How does substitution help in verifying the equality of algebraic expressions?

### Tip:
Always simplify each side of an equation separately before comparing them for equality.

Explore the steps to identify false algebraic expressions, including simplifying exponents, analyzing proportions, and applying the distributive property. This problem is ideal for students in Grades 9-11 and covers essential algebraic concepts.

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