The real solution of the following equation lies between (a, b) where a and
b are the smallest integer such that a < x < b.
(x − 1/x)^1/2 + (1 −1/x)^1/2 = x.

Then find the value of a + b.

We need to find the interval \((a, b)\) such that the real solution \(x\) of the equation lies between \(a\) and \(b\). The equation is given by:

\[
\sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}} = x
\]

Let's approach the problem step by step.

### Step 1: Analyze the function and the given equation
First, note that for \(\sqrt{x - \frac{1}{x}}\) and \(\sqrt{1 - \frac{1}{x}}\) to be real, both \(x - \frac{1}{x} \geq 0\) and \(1 - \frac{1}{x} \geq 0\) must hold.

1. **Condition \(x - \frac{1}{x} \geq 0\):**
   - This implies \(x \geq 1\).

2. **Condition \(1 - \frac{1}{x} \geq 0\):**
   - This implies \(x \geq 1\).

Thus, \(x \geq 1\) is necessary for the equation to have real solutions.

### Step 2: Evaluate the function near potential points of interest
Let's test specific values of \(x\) to see where the solution might lie:

1. **For \(x = 1\):**

   \[
   \sqrt{1 - \frac{1}{1}} + \sqrt{1 - \frac{1}{1}} = \sqrt{0} + \sqrt{0} = 0 \quad \text{(Not equal to 1, so \(x = 1\) is not a solution)}
   \]

2. **For \(x = 2\):**

   \[
   \sqrt{2 - \frac{1}{2}} + \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{3}{2}} + \sqrt{\frac{1}{2}} \approx 1.22 + 0.71 = 1.93 \quad \text{(Not equal to 2)}
   \]

3. **For \(x = 3\):**

   \[
   \sqrt{3 - \frac{1}{3}} + \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{8}{3}} + \sqrt{\frac{2}{3}} \approx 1.63 + 0.82 = 2.45 \quad \text{(Not equal to 3)}
   \]

4. **For \(x = 4\):**

   \[
   \sqrt{4 - \frac{1}{4}} + \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{15}{4}} + \sqrt{\frac{3}{4}} \approx 1.93 + 0.87 = 2.80 \quad \text{(Not equal to 4)}
   \]

5. **For \(x = 5\):**

   \[
   \sqrt{5 - \frac{1}{5}} + \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{24}{5}} + \sqrt{\frac{4}{5}} \approx 2.19 + 0.89 = 3.08 \quad \text{(Not equal to 5)}
   \]

Since none of these exact values satisfy the equation, and the function seems to increase slowly, we suspect the solution \(x\) lies between 3 and 4.

### Step 3: Conclude the interval \((a, b)\)
Given that the solution lies between 3 and 4:

\[
a = 3, \quad b = 4
\]

### Step 4: Calculate \(a + b\)
\[
a + b = 3 + 4 = \boxed{7}
\]

This is the final answer.

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Would you like further details or have any questions?

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**Here are 5 related questions:**
1. How can we prove that \(x = 3.5\) is closer to the solution than \(x = 3\) or \(x = 4\)?
2. What other methods can we use to approximate the solution of nonlinear equations?
3. How does the behavior of the function change as \(x\) increases beyond 4?
4. What is the derivative of the function \(f(x) = \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}} - x\)?
5. Can this problem be solved using numerical methods like Newton's method?

**Tip:** For finding roots of nonlinear equations, numerical methods like the bisection method or Newton's method are often more practical than algebraic manipulation.

Discover how to find the interval (a, b) where the real solution of the equation lies between, using a detailed step-by-step analysis. Explore the conditions and evaluations for finding a + b. Suitable for advanced high school students and enthusiasts of mathematical problem-solving.

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