Simplify the sum or difference. State any restrictions on the variable. 3x/x^2-4 + 6/x+2
To simplify the sum or difference, we need a common denominator. Factoring x^2-4 gives us (x+2)(x-2), so we can write:
3x/(x+2)(x-2) + 6/(x+2)
To add these fractions, we need a common denominator of (x+2)(x-2). We can find the equivalent fraction for 6/(x+2) by multiplying the numerator and denominator by (x-2):
3x/(x+2)(x-2) + 6(x-2)/(x+2)(x-2)
Combining the fractions gives:
(3x+6x-12)/(x+2)(x-2) = (9x-12)/(x+2)(x-2)
The restrictions on the variable come from the fact that we cannot divide by zero. Therefore, x cannot be equal to -2 or 2.
To simplify the given sum, we need to find a common denominator for the fractions.
The denominators are x^2 - 4 and x + 2. We can factor x^2 - 4 as (x - 2)(x + 2).
Now, the common denominator is (x + 2)(x - 2).
Rewriting the fractions with the common denominator, we get:
3x/(x^2 - 4) + 6/(x + 2) = 3x/[(x + 2)(x - 2)] + 6/(x + 2).
Next, we need to add the fractions:
[3x + 6(x - 2)]/[(x + 2)(x - 2)].
Simplifying the numerator:
[3x + 6x - 12]/[(x + 2)(x - 2)].
Combining like terms:
[9x - 12]/[(x + 2)(x - 2)].
This is the simplified expression.
Restrictions on the variable:
To avoid division by zero, the denominator (x + 2)(x - 2) cannot be equal to zero. Thus, the variable must satisfy the restrictions x ≠ -2 and x ≠ 2.