If ` x_(1) x_(2) x_(3) = 4 ( 4 + x_(1) + x_(2)+ x_(3))` then what is the value of `[(1)/(( 2 + x_(1))] + [(1)/(( 2 + x_(2)) ] + (1)/( (2 + x_(3))]` ?

Options

A. 1
B. `(1)/(2)`
C. 2
D. `(1)/(3)`

Answer

To solve the problem, we start with the given equation:

\[ x_1 x_2 x_3 = 4(4 + x_1 + x_2 + x_3) \]

### Step 1: Assume values for \( x_1, x_2, x_3 \)

Let's assume \( x_1 = x_2 = x_3 = 4 \). This is a reasonable assumption because the equation is symmetric in \( x_1, x_2, x_3 \).

### Step 2: Substitute the assumed values into the equation

Now, substituting \( x_1 = 4, x_2 = 4, x_3 = 4 \) into the left-hand side:

\[ x_1 x_2 x_3 = 4 \cdot 4 \cdot 4 = 64 \]

Now, substituting into the right-hand side:

\[ 4(4 + x_1 + x_2 + x_3) = 4(4 + 4 + 4 + 4) = 4(16) = 64 \]

### Step 3: Confirm that both sides are equal

Both sides are equal, confirming that our assumption is valid:

\[ 64 = 64 \]

### Step 4: Calculate the desired expression

Now we need to find the value of:

\[ \frac{1}{2 + x_1} + \frac{1}{2 + x_2} + \frac{1}{2 + x_3} \]

Substituting \( x_1 = 4, x_2 = 4, x_3 = 4 \):

\[ \frac{1}{2 + 4} + \frac{1}{2 + 4} + \frac{1}{2 + 4} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \]

### Step 5: Combine the fractions

Adding these fractions together:

\[ \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{1}{2} \]

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