Simplify. Express your answer as a single fraction in simplest form.

Question

10
2p–6
–
p
6p–18

Answer 1

To simplify the expression \(\frac{10}{2p - 6} - \frac{p}{6p - 18}\), we first factor the denominators.

1. Factor the denominators:

   • The first denominator: \[ 2p - 6 = 2(p - 3) \]
   • The second denominator: \[ 6p - 18 = 6(p - 3) \]

2. Rewrite the expression with factored denominators: \[ \frac{10}{2(p - 3)} - \frac{p}{6(p - 3)} \]

3. Find a common denominator: The common denominator is \(6(p - 3)\).

4. Rewrite each fraction with the common denominator:

   • For the first fraction: \[ \frac{10}{2(p - 3)} = \frac{10 \cdot 3}{2(p - 3) \cdot 3} = \frac{30}{6(p - 3)} \]
   • The second fraction is already in terms of the common denominator: \[ \frac{p}{6(p - 3)} \]

5. Combine the fractions: \[ \frac{30}{6(p - 3)} - \frac{p}{6(p - 3)} = \frac{30 - p}{6(p - 3)} \]

6. Simplify the final fraction if possible: The expression \(\frac{30 - p}{6(p - 3)}\) does not simplify further because \(30 - p\) does not factor.

Thus, the answer in simplest form is: \[ \frac{30 - p}{6(p - 3)} \]