Question

simplify. express your answer as a single fraction in simplest form.\\(\frac{2w + 6}{w + 6} - \frac{5}{w - 1}\\)

Explanation:

Step1: Find common denominator

The common denominator of \(w + 6\) and \(w - 1\) is \((w + 6)(w - 1)\). Rewrite each fraction with this common denominator:
\(\frac{(2w + 6)(w - 1)}{(w + 6)(w - 1)}-\frac{5(w + 6)}{(w + 6)(w - 1)}\)

Step2: Expand numerators

Expand the numerators:
\((2w + 6)(w - 1)=2w^2-2w + 6w-6 = 2w^2 + 4w-6\)
\(5(w + 6)=5w + 30\)
So the expression becomes \(\frac{2w^2 + 4w-6-(5w + 30)}{(w + 6)(w - 1)}\)

Step3: Simplify numerator

Simplify the numerator by distributing the negative sign and combining like terms:
\(2w^2+4w - 6-5w - 30=2w^2 - w-36\)
Now we have \(\frac{2w^2 - w - 36}{(w + 6)(w - 1)}\)

Step4: Factor numerator (if possible)

Factor the quadratic \(2w^2 - w - 36\). We need two numbers that multiply to \(2\times(-36)=-72\) and add to \(-1\). The numbers are \(-9\) and \(8\).
Rewrite the middle term: \(2w^2+8w - 9w - 36\)
Factor by grouping: \(2w(w + 4)-9(w + 4)=(2w - 9)(w + 4)\)
So the simplified fraction is \(\frac{(2w - 9)(w + 4)}{(w + 6)(w - 1)}\) (we can check that the numerator and denominator have no common factors, so this is the simplest form)

Answer:

\(\frac{(2w - 9)(w + 4)}{(w + 6)(w - 1)}\) (or expanded numerator \(\frac{2w^2 - w - 36}{(w + 6)(w - 1)}\) is also correct, but factored form is more simplified)