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To solve this problem, we need to analyze the two given equations involving square roots:

1. \[
\sqrt{x - 1} + \sqrt{y - 1} + \sqrt{z - 1} = a - 1
\]
2. \[
\sqrt{x + 1} + \sqrt{y + 1} + \sqrt{z + 1} = a + 1
\]

### Step 1: Analyze the constraints
- The square roots are defined only when their arguments are non-negative, meaning:
    - \( x - 1 \geq 0 \implies x \geq 1 \)
    - \( y - 1 \geq 0 \implies y \geq 1 \)
    - \( z - 1 \geq 0 \implies z \geq 1 \)
    - Similarly, \( x + 1 \geq 0 \), \( y + 1 \geq 0 \), \( z + 1 \geq 0 \), which hold true for all real values of \( x \), \( y \), and \( z \).

### Step 2: Find minimum values for \( a \)
To determine the minimum value of \( a \), let’s consider the smallest possible values for \( \sqrt{x - 1} \), \( \sqrt{y - 1} \), and \( \sqrt{z - 1} \).

1. When \( x = y = z = 1 \):
   - \(\sqrt{x - 1} = \sqrt{y - 1} = \sqrt{z - 1} = 0\).
   - Substituting into the first equation:
     \[
     0 + 0 + 0 = a - 1 \implies a - 1 = 0 \implies a = 1.
     \]

2. Now, consider the second equation:
   - When \( x = y = z = 1 \):
     \[
     \sqrt{x + 1} = \sqrt{y + 1} = \sqrt{z + 1} = \sqrt{2}.
     \]
   - Substituting into the second equation:
     \[
     \sqrt{2} + \sqrt{2} + \sqrt{2} = a + 1 \implies 3\sqrt{2} = a + 1 \implies a = 3\sqrt{2} - 1.
     \]

Since we want the minimum value of \( a \) such that the system has real solutions, we take the larger of the two values derived. Evaluating \( 3\sqrt{2} \approx 4.24 \), so \( a = 3\sqrt{2} - 1 \approx 3.24 \).

This value isn't an option in the given choices, so the correct answer is **D: None of the foregoing options**.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do square root equations affect the domain of possible solutions?
2. Why do we consider the minimum values of \( x \), \( y \), and \( z \) in this type of problem?
3. What would change in the solution if \( \sqrt{x + 1} \) were replaced by \( \sqrt{x + 2} \)?
4. How does the value of \( a \) influence the range of possible solutions for \( x \), \( y \), and \( z \)?
5. What methods can be used to find the range of values for square root equations systematically?

**Tip:** When dealing with square root equations, always check the domain to ensure the roots are defined for the given values.

Find minimal value of a ∈ ℝ such that system √(x - 1) + √(y - 1) + √(z - 1) = a - 1 and √(x + 1) + √(y + 1) + √(z + 1) = a + 1 has solution in set of real numbers.

Learn how to determine the minimum value of a for which the system of square root equations √(x - 1) + √(y - 1) + √(z - 1) = a - 1 and √(x + 1) + √(y + 1) + √(z + 1) = a + 1 has a solution in real numbers. This problem explores key concepts in algebra and systems of equations, suitable for students in Grades 11-12.

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