Question

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

Answer 1

Let's first find the restrictions:

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

`(3x - 2)/(x + 3) + 7/((x - 4)(x + 3))`

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.

`x + 3 = 0 and x - 4 = 0`

`x = -3 and x = 4`

Therefore, `x!= -3, 4`
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Now, let's simplify:

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

Since both expressions already has `(x + 3)` as a factor in the denominator, and only one has `(x - 4)`, we must multiply the expression to the left by `(x - 4)` to make it equivalent to the one on the right.

`((3x - 2)(x - 4))/((x + 3)(x - 4)) + 7/((x + 3)(x - 4))`

`=(3x^2 - 14x + 8 + 7)/((x + 3)(x - 4))`

`=(3x^2 - 14x + 15)/((x + 3)(x - 4))`

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!