Final answer:
To add two rational expressions with unlike denominators and find a sum of (x+1)/(2(x-3)), we need to find a common denominator for both expressions. Upon examination, option d provides expressions that can be made equivalent to the desired sum when simplified.
Step-by-step explanation:
When working with rational expressions, finding a common denominator is essential for adding fractions with unlike denominators. To achieve a common denominator, one can multiply each fraction by a form of 1 that incorporates the missing factors from the opposing fraction's denominator. This process ensures that both fractions have the same denominator, allowing the numerators to be added directly. In the given problem, we look for a pair of rational expressions that sum to (x+1)/(2(x-3)).
Let's analyze option d) (x-3)/(2x) and (x+1)/(2x-6). These fractions have similar but not identical denominators. We can see that 2x and 2x-6 will never be equal unless x is infinite, which is not possible in this context. However, if we factor out a 2 from the second denominator, we see that it becomes (x+1)/(2(x-3)), which is the desired sum. To confirm, we simplify the first fraction's denominator which can be done by multiplying by (x-3)/(x-3) to get a common denominator of 2(x-3). So the correct expressions are (x-3)/(2(x-3)) and (x+1)/(2(x-3)).