Let's simplify the given expression step by step to find which option is equivalent to the difference shown.
The expression given is
[tex]\[
\frac{5x + 1}{5x} - \frac{4x + 1}{4x}
\][/tex]
First, we simplify each fraction separately:
[tex]\[
\frac{5x + 1}{5x} = \frac{5x}{5x} + \frac{1}{5x} = 1 + \frac{1}{5x}
\][/tex]
[tex]\[
\frac{4x + 1}{4x} = \frac{4x}{4x} + \frac{1}{4x} = 1 + \frac{1}{4x}
\][/tex]
Now we subtract these two expressions:
[tex]\[
\left(1 + \frac{1}{5x}\right) - \left(1 + \frac{1}{4x}\right)
\][/tex]
Simplify the subtraction inside the parentheses:
[tex]\[
1 + \frac{1}{5x} - 1 - \frac{1}{4x} = \frac{1}{5x} - \frac{1}{4x}
\][/tex]
Now, we need to combine these two fractions. To do that, we find the common denominator, which is [tex]\(20x\)[/tex]:
[tex]\[
\frac{1}{5x} = \frac{4}{20x}
\][/tex]
[tex]\[
\frac{1}{4x} = \frac{5}{20x}
\][/tex]
Now, subtract these fractions:
[tex]\[
\frac{4}{20x} - \frac{5}{20x} = \frac{4 - 5}{20x} = \frac{-1}{20x}
\][/tex]
Thus, the simplified form of the original expression is
[tex]\[
-\frac{1}{20x}
\][/tex]
Hence, the correct answer is:
A. [tex]\(-\frac{1}{20 x}\)[/tex]
