simplify the sum and state and restrictions on the variables.
(x-2)/(x+3) + (10x)/(x^2-9)
To simplify the sum, we need to find a common denominator.
The first fraction can be rewritten as (x-2)/(x+3) * (x-3)/(x-3) = (x^2-5x+6)/(x^2-9)
The second fraction is already in the form with common denominator.
Now we can add the fractions:
(x^2-5x+6)/(x^2-9) + (10x)/(x^2-9)
= (x^2-5x+6+10x)/(x^2-9)
= (x^2+5x+6)/(x^2-9)
So the simplified sum is (x^2+5x+6)/(x^2-9).
The restriction on the variables are such that the denominator cannot be zero, since division by zero is undefined. This means that x cannot equal 3 or -3.
the simplified answer should be (x+2)/(x-3)
Apologies for the mistake in the simplification earlier. Let's simplify the sum again:
(x-2)/(x+3) + (10x)/(x^2-9)
First, factor the denominator of the second fraction:
x^2 - 9 = (x + 3)(x - 3)
Now, rewrite the sum with a common denominator:
[(x-2)(x-3) + 10x] / (x+3)(x-3)
= (x^2 - 5x + 6 + 10x) / (x^2 - 9)
= (x^2 + 5x + 6) / (x^2 - 9)
Now, factor the numerator:
x^2 + 5x + 6 = (x + 2)(x + 3)
Thus, the simplified sum is (x + 2)/(x - 3).
The restriction on the variables is still that x cannot equal 3 or -3, as those values would make the denominator zero.