To solve the equation \( \frac{x + 1}{x + 7} = \frac{7}{6} \), we can use cross-multiplication:
\[ 6(x + 1) = 7(x + 7) \]
Now, let's distribute on both sides:
\[ 6x + 6 = 7x + 49 \]
Next, we'll isolate the variable \( x \). Start by getting all \( x \) terms on one side and constant terms on the other side:
\[ 6 = 7x - 6x + 49 \]
This simplifies to:
\[ 6 = x + 49 \]
Now, subtract 49 from both sides:
\[ 6 - 49 = x \]
This gives us:
\[ x = -43 \]
So the solution to the equation is:
\[ \boxed{-43} \]