Simplify each sum or difference. State any restrictions on the variable.
-3x/x^2-9+4/2x-6
To simplify the given expression (-3x/x^2-9)+(4/2x-6), we can start by factoring the denominators of each fraction.
The denominator of the first fraction, x^2 - 9, is a difference of squares and can be factored as (x - 3)(x + 3).
The denominator of the second fraction, 2x - 6, can be factored out the greatest common factor, which is 2, resulting in 2(x - 3).
Now, we rewrite the expression with the factored denominators:
-3x/(x - 3)(x + 3) + 4/[2(x - 3)]
To find the least common denominator (LCD) for these fractions, we need to take the product of the distinct factors of both denominators. In this case, the LCD is (x - 3)(x + 3).
Next, we need to rewrite the fractions with this common denominator:
[-3x/(x - 3)(x + 3)] * [(x + 3)/(x + 3)] + 4/[2(x - 3)] * [(x + 3)/(x + 3)]
Distributing and simplifying the numerators, we get:
(-3x(x + 3) + 4(x + 3)) / [(x - 3)(x + 3)]
Multiplying out the numerators, we have:
(-3x^2 - 9x + 4x + 12) / [(x - 3)(x + 3)]
Combining like terms in the numerator:
(-3x^2 - 5x + 12) / [(x - 3)(x + 3)]
So, the simplified expression is (-3x^2 - 5x + 12) / [(x - 3)(x + 3)].
As for restrictions on the variable, we need to check if there are any values of x that would make the denominators equal to zero. In this case, we have (x - 3)(x + 3) as the denominator. Thus, the restrictions on x are x โ 3 and x โ -3.