Sure, let's solve this problem step-by-step.
First, there are two rational expressions to consider:
1. (t + 2) / (3t - 9)
2. (3 - 2t) / (5t - 15)
Here's how we can address the subtraction.
First step is to simplify the expressions by factoring out common factors, if available. Looking at both of the denominators, we can see that 3t-9 can be re-written as 3(t-3), and 5t-15 can be re-written as 5(t-3). This leaves us with:
1. (t + 2) / 3(t - 3)
2. (3 - 2t) / 5(t - 3)
Before subtracting these expressions, they need to have the same denominator. Since they both share a factor of t-3, we can use 15(t-3) as a common denominator. To do this, multiply the first expression by 5/5 and the second by 3/3 so that they have the same denominator:
1. 5(t + 2) / 15(t - 3)
2. 3(3 - 2t) / 15(t - 3)
Now you can combine these two fractions since they have the same denominator:
1. [5(t + 2) + 3(3 - 2t)] / 15(t - 3)
Distribute and combine like terms in the numerator:
1. (5t + 10 + 9 - 6t) / 15(t - 3)
2. (19 - t) / 15(t - 3)
As you can see, subtracting the two rational expressions gives us: (19 - t) / 15(t - 3)
This is your answer.