Let's solve the equation step by step:

The given equation is:

\[
\frac{x - 4}{4x} + 1 = \frac{x + 4}{x}
\]

### Step 1: Eliminate the fractions
We can multiply both sides of the equation by the least common denominator (LCD). The LCD of \(4x\) and \(x\) is \(4x\). So, we multiply both sides by \(4x\):

\[
4x \left( \frac{x - 4}{4x} + 1 \right) = 4x \left( \frac{x + 4}{x} \right)
\]

This simplifies to:

\[
(x - 4) + 4x = 4(x + 4)
\]

### Step 2: Expand both sides
Now, expand both sides:

\[
x - 4 + 4x = 4x + 16
\]

Simplify the left side:

\[
5x - 4 = 4x + 16
\]

### Step 3: Isolate \(x\)
Move all the \(x\)-terms to one side and constants to the other side:

\[
5x - 4x = 16 + 4
\]

Simplify:

\[
x = 20
\]

### Solution:
The solution is \(x = 20\).

Would you like to see the details of each step or have any questions?

Here are some related questions:
1. What happens if we multiply both sides by different LCDs?
2. How would the solution change if we had a different constant?
3. Can you verify the solution by substituting \(x = 20\) into the original equation?
4. How does isolating \(x\) work when dealing with fractions?
5. Can we rearrange the equation in another way to solve it?

**Tip**: Always check for extraneous solutions when working with equations involving fractions.

This page provides a step-by-step solution for solving the rational equation x - 4 / 4x + 1 = x + 4 / x by eliminating fractions using the Least Common Denominator (LCD) and solving for x. Ideal for students in Grades 9-10.

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