12. Write the following as a single fraction in its simplest form - Junior Cycle Mathematics - Question 12 - 2021

To combine the fractions $\frac{2}{n - 3}$ and $\frac{5}{2n + 5}$, we first need a common denominator. The common denominator is $(n - 3)(2n + 5)$. Now, express each fraction with this common denominator:

$\frac{2(2n + 5)}{(n - 3)(2n + 5)} - \frac{5(n - 3)}{(n - 3)(2n + 5)}$

Now we can write them as:

$\frac{2(2n + 5) - 5(n - 3)}{(n - 3)(2n + 5)}$

Next, simplify the numerator:

1. Expand:

   • $2(2n + 5) = 4n + 10$
   • $-5(n - 3) = -5n + 15$

2. Combine: $4n + 10 - 5n + 15 = -n + 25$

Thus, we have:

$\frac{-n + 25}{(n - 3)(2n + 5)}$

This is the single fraction in its simplest form.