Let's solve the equation:
[tex]\[
-\frac{7}{w-6} = \frac{1}{4w - 24} + 1
\][/tex]
First, let's simplify the right-hand side:
[tex]\[
\frac{1}{4w - 24} + 1
\][/tex]
We can observe that:
[tex]\[
4w - 24 = 4(w - 6)
\][/tex]
Therefore:
[tex]\[
\frac{1}{4w - 24} = \frac{1}{4(w - 6)}
\][/tex]
So the original equation becomes:
[tex]\[
-\frac{7}{w-6} = \frac{1}{4(w-6)} + 1
\][/tex]
Next, let's find a common denominator for the terms on the right:
[tex]\[
\frac{1}{4(w-6)} + 1 = \frac{1}{4(w-6)} + \frac{4(w-6)}{4(w-6)} = \frac{1 + 4(w-6)}{4(w-6)}
\][/tex]
Simplify the expression:
[tex]\[
\frac{1 + 4w - 24}{4(w-6)} = \frac{1 + 4w - 24}{4(w-6)} = \frac{4w - 23}{4(w-6)}
\][/tex]
So the equation becomes:
[tex]\[
-\frac{7}{w-6} = \frac{4w - 23}{4(w-6)}
\][/tex]
To clear the denominators, multiply both sides by [tex]\(4(w - 6)\)[/tex]:
[tex]\[
-7 \cdot 4 = 4w - 23
\][/tex]
Simplify and solve for [tex]\(w\)[/tex]:
[tex]\[
-28 = 4w - 23
\][/tex]
Add 23 to both sides:
[tex]\[
-28 + 23 = 4w
\][/tex]
[tex]\[
-5 = 4w
\][/tex]
Divide both sides by 4:
[tex]\[
w = -\frac{5}{4}
\][/tex]
So the solution to the equation is:
[tex]\[
w = -\frac{5}{4}
\][/tex]
[tex]\[
\boxed{-\frac{5}{4}}
\][/tex]
