Final answer:
To add the rational expressions (x-3)/6 and (x+5)/7, we equate the denominators to 42 by multiplying by appropriate factors, combine the numerators and simplify to get the final answer (13x + 9) / 42.
Step-by-step explanation:
To add the rational expressions (x-3)/6 and (x+5)/7, we need to find a common denominator. The least common denominator (LCD) of 6 and 7 is 42 since 6 and 7 are both prime numbers and have no common factors other than 1. We then express each fraction with the denominator of 42.
For the first fraction: ((x-3)/6) × (7/7) = (7x - 21) / 42.
For the second fraction: ((x+5)/7) × (6/6) = (6x + 30) / 42.
Now both fractions have the same denominator, so we can combine them.
The sum is: (7x - 21 + 6x + 30) / 42 = (13x + 9) / 42.
We can check to see if this expression can be simplified further, but since 13x + 9 has no common factors with 42, this is the simplified expression.