`(2)/(p)-(1)/(2p)=(p^(2)+1)/(p^(2)+1)`
In the equation above, what is the value of `(1)/(p)`?

Options

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

Answer

To solve the equation

\[ \frac{2}{p} - \frac{1}{2p} = \frac{p^2 + 1}{p^2 + 1} \]

we can simplify the right-hand side first. Since \( \frac{p^2 + 1}{p^2 + 1} = 1 \) (as long as \( p^2 + 1 \neq 0 \)), we can rewrite the equation as:

\[ \frac{2}{p} - \frac{1}{2p} = 1 \]

Next, we need to find a common denominator for the left-hand side. The common denominator for \( p \) and \( 2p \) is \( 2p \). Therefore, we can rewrite the left-hand side:

\[ \frac{2 \cdot 2}{2p} - \frac{1}{2p} = \frac{4}{2p} - \frac{1}{2p} = \frac{4 - 1}{2p} = \frac{3}{2p} \]

Now we have:

\[ \frac{3}{2p} = 1 \]

To eliminate the fraction, we can cross-multiply:

\[ 3 = 2p \]

Now, we can solve for \( p \):

\[ p = \frac{3}{2} \]

Next, we need to find the value of \( \frac{1}{p} \):

\[ \frac{1}{p} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

Thus, the value of \( \frac{1}{p} \) is:

\[ \frac{2}{3} \]

### Summary of Steps:
1. Simplify the right-hand side of the equation.
2. Find a common denominator for the left-hand side.
3. Combine the fractions on the left-hand side.
4. Set the simplified equation equal to 1.
5. Cross-multiply to eliminate the fraction.
6. Solve for \( p \).
7. Find \( \frac{1}{p} \).