Simplify the expression completely.
1
x + 5
 − 5
25
x
 − x

To simplify the expression 

\[
\frac{1}{x + 5} - \frac{5}{25x} - x,
\]

let's follow these steps:

### Step 1: Simplify the second fraction
The second term \(\frac{5}{25x}\) can be simplified by dividing the numerator and the denominator by 5:

\[
\frac{5}{25x} = \frac{1}{5x}.
\]

### Step 2: Rewrite the expression
Now the expression becomes:

\[
\frac{1}{x + 5} - \frac{1}{5x} - x.
\]

### Step 3: Find a common denominator
The common denominator for the fractions \(\frac{1}{x + 5}\) and \(\frac{1}{5x}\) is \((x + 5)(5x)\). Rewrite both fractions with this common denominator:

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

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

### Step 4: Combine the fractions
Now, subtract the second fraction from the first:

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

Simplify the numerator:

\[
5x - (x + 5) = 5x - x - 5 = 4x - 5.
\]

So, the fraction simplifies to:

\[
\frac{4x - 5}{5x(x + 5)}.
\]

### Step 5: Combine the simplified fraction with the term \(-x\)
Now, include the \(-x\) term:

\[
\frac{4x - 5}{5x(x + 5)} - x.
\]

### Step 6

Simplify the expression completely.
1/(x + 5) − 5/(25x) − x

Learn how to simplify the rational expression 1/(x + 5) − 5/(25x) − x with a step-by-step explanation. This problem demonstrates the use of common denominators and fraction subtraction, suitable for students in Grades 9-11.

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