If `(x + 5)(1/x + 1/5)=4` then x=

Options

A. `1//5`
B. `1//2`
C. `1`
D. `5`

Answer

To solve the equation \((x + 5)\left(\frac{1}{x} + \frac{1}{5}\right) = 4\), we will follow these steps:

### Step 1: Expand the left-hand side of the equation
We start by expanding the expression on the left-hand side:

\[ (x + 5)\left(\frac{1}{x} + \frac{1}{5}\right) = (x + 5)\left(\frac{5 + x}{5x}\right) \]

### Step 2: Distribute the terms
Now, we can distribute \(x + 5\) across \(\frac{5 + x}{5x}\):

\[ = \frac{(x + 5)(5 + x)}{5x} \]

### Step 3: Simplify the expression
Now, we can simplify the numerator:

\[ = \frac{x^2 + 5x + 5x + 25}{5x} = \frac{x^2 + 10x + 25}{5x} \]

### Step 4: Set the equation equal to 4
Now we set the expression equal to 4:

\[ \frac{x^2 + 10x + 25}{5x} = 4 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ x^2 + 10x + 25 = 20x \]

### Step 6: Rearrange the equation
Now, we rearrange the equation:

\[ x^2 + 10x + 25 - 20x = 0 \]
\[ x^2 - 10x + 25 = 0 \]

### Step 7: Factor the quadratic equation
Next, we factor the quadratic equation:

\[ (x - 5)(x - 5) = 0 \]

### Step 8: Solve for x
Setting each factor equal to zero gives us:

\[ x - 5 = 0 \implies x = 5 \]

### Final Answer
Thus, the solution to the equation is:

\[ \boxed{5} \]