Answer
The solution is $p = \frac{1}{2}$.
Work Step by Step
The first thing we want to do is get rid of the fractions. We do this by first finding the least common denominator (LCD) for all the terms.
We find the LCD by taking the highest power of each factor in the denominators of the fractions:
LCD = $2p$
Now, we will multiply each numerator by the factor its denominator is missing from the LCD:
$3(2p) + 5 = 4(2)$
Multiply to simplify:
$6p + 5 = 8$
Collect constants on the right side of the equation:
$6p = 3$
Divide both sides by $6$:
$p = \frac{3}{6}$
Simplify the fraction:
$p = \frac{1}{2}$
The solution is $p = \frac{1}{2}$.
To check our answers, we plug the values into the original equation.
$3 + \frac{5}{2(\frac{1}{2})} = \frac{4}{\frac{1}{2}}$
Multiply to simplify:
$3 + \frac{5}{1} = \frac{4}{\frac{1}{2}}$
Simplify the fractions by multiplying by the inverse of the denominator:
$3 + 5 = \frac{8}{1}$
Simplify the fractions:
$3 + 5 = 8$
Add:
$8 = 8$
This solution is correct.
