Question

solve for all values of x(multiple are allowed)

1/x+7 + 1=- x/x+7

Answer 1

To solve the equation

\[
\frac{1}{x+7} + 1 = -\frac{x}{x+7},
\]

first, let's simplify the equation. We can express 1 as \(\frac{x+7}{x+7}\) to have a common denominator:

\[
\frac{1}{x+7} + \frac{x+7}{x+7} = -\frac{x}{x+7}.
\]

Now, combining the left side gives us:

\[
\frac{1 + (x+7)}{x+7} = \frac{x + 8}{x + 7}.
\]

Now, the equation looks like this:

\[
\frac{x + 8}{x + 7} = -\frac{x}{x + 7}.
\]

Since both sides have the common denominator \(x + 7\), and assuming \(x + 7 \neq 0\) (which we will check later), we can cross-multiply:

\[
x + 8 = -x.
\]

Next, solve for \(x\):

\[
x + x + 8 = 0 \\
2x + 8 = 0 \\
2x = -8 \\
x = -4.
\]

Now, we need to ensure this solution does not make the denominator zero. We check:

\[
x + 7 \neq 0 \implies -4 + 7 \neq 0 \implies 3 \neq 0.
\]

Since \(x + 7\) does not equal zero when \(x = -4\), the solution is valid.

Thus, the solution is:

\[
\boxed{-4}.
\]