How do you find the restrictions and simplify #(3x-2)/(x+3)+7/(x^2-x-12)#?

Let's first find the restrictions:

This can be done by factoring the denominator of the expression.

A restriction in a rational expression occurs when the denominator equals 0, since division by 0 in mathematics is undefined.

Therefore, we must now set the factors in the denominator to 0 and solve for x. These will be our restrictions.

This can be done by placing everything on a common denominator.

The trinomial in the numerator is factorable. Always factor it when possible to see if anything can be simplified. You will be docked marks if you don't simplify fully. I factored this one and nothing needs to be eliminated. This is in simplest form.

Hopefully this helps!

To find the restrictions, we need to identify any values of x that would make the denominator(s) equal to zero. In this case, the first denominator is (x+3) and the second denominator is (x^2-x-12).

Setting the first denominator equal to zero, we have x+3=0. Solving for x, we find x=-3.

For the second denominator, we can factor it as (x-4)(x+3). Setting this equal to zero, we have (x-4)(x+3)=0. Solving for x, we find x=4 and x=-3.

Therefore, the restrictions are x=-3 and x=4.

To simplify the expression (3x-2)/(x+3)+7/(x^2-x-12), we first need to find a common denominator for the two fractions. The common denominator is (x+3)(x-4).

Multiplying the first fraction by (x-4)/(x-4) and the second fraction by (x+3)/(x+3), we get (3x-2)(x-4)/[(x+3)(x-4)] + 7(x+3)/[(x+3)(x-4)].

Combining the fractions, we have [(3x-2)(x-4) + 7(x+3)]/[(x+3)(x-4)].

Expanding and simplifying the numerator, we get (3x^2 - 14x - 8 + 7x + 21)/[(x+3)(x-4)].

Combining like terms, we have (3x^2 - 7x + 13)/[(x+3)(x-4)].

Therefore, the simplified expression is (3x^2 - 7x + 13)/[(x+3)(x-4)].