Final answer:
To simplify the rational expression (4-x)/(3x²-48), factor both the numerator and the denominator to cancel out common terms, resulting in the simplified expression -1/(3(x+4)), with restrictions on x being not equal to 4 or -4.
Step-by-step explanation:
To simplify the rational expression into lowest terms for the given expression (4-x)/(3x²-48), we must factor both the numerator and the denominator. First, the numerator is already simplified, but we can rewrite it as (-(x-4)) to reveal common factors later on. For the denominator, we factor out the greatest common factor, which is 3, and we get 3(x²-16). The term (x²-16) is a difference of squares that can be factored further into (x-4)(x+4).
Now the expression becomes:
(-(x-4))/(3(x-4)(x+4))
Then, we can cancel the (x-4) terms out. Thus, the expression simplified is:
-1/(3(x+4))
Remember to consider any restrictions on the variable x, which in this case cannot be 4 or -4 since these would make the original denominator equal to zero.