Question

solve for all values of x

x/x+2 - 5=- 2/x+2

Answer 1

To solve the equation

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

let's first combine the terms on the left-hand side. To get started, we can first eliminate the fraction by isolating the fractions on one side:

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

Now we can combine the left side:

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

This simplifies to:

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

Now, since \(x + 2 \neq 0\), we can multiply both sides by \(x + 2\):

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

Now we expand the right side:

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

Next, let's move all terms with \(x\) to one side and constant terms to the other side:

\[
2 - 10 = 5x - x,
\]

which simplifies to:

\[
-8 = 4x.
\]

Now, divide both sides by 4:

\[
x = -2.
\]

Now, we need to verify that this solution does not make any denominator zero in the original equation.

Substituting \(x = -2\) into the original denominators \(x + 2\):

\[
-2 + 2 = 0.
\]

Since this value makes the denominator zero, \(x = -2\) is not a valid solution for the equation.

Thus, we conclude that there are **no valid solutions** for the given equation.