Question

solve for all values of x
there can be two different values
x/x+2 - 5=- 2/x+2

Answer 1

To solve the equation

\[
\frac{x}{x+2} - 5 = -\frac{2}{x+2},
\]

first, we'll rewrite it in a more manageable form.

1. Move \(-\frac{2}{x+2}\) to the left side:

\[
\frac{x}{x+2} + \frac{2}{x+2} - 5 = 0.
\]

2. Combine the fractions on the left side:

\[
\frac{x + 2}{x+2} - 5 = 0.
\]

This simplifies to:

\[
\frac{x + 2}{x + 2} - 5 = 0,
\]

or

\[
1 - 5 = 0.
\]

3. Rearranging this, we get:

\[
1 - 5 = -4.
\]

Now, let's multiply both sides by \(x + 2\) (keeping in mind that \(x + 2 \neq 0\) to avoid division by zero):

\[
x + 2 - 5(x + 2) = -2.
\]

Expanding gives:

\[
x + 2 - 5x - 10 = -2.
\]

Combine like terms:

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

4. Add \(8\) to both sides:

\[
-4x = 6.
\]

5. Now, divide by \(-4\):

\[
x = -\frac{3}{2}.
\]

This is one potential solution. We can also check for \( x = -2 \) since it makes the fractions undefined, so we exclude it as a possible solution.

6. Thus, the solution for the equation is:

\[
\boxed{-\frac{3}{2}}.
\]

To check if there are any more solutions, we can revisit our steps, but it looks like the algebra provided above leads us to only one valid solution for \(x\), which is \( -\frac{3}{2} \).