Solve for the rational equation x-2 over x +3  separate then - 1=3 over x +2

To solve the equation 

\[
\frac{x - 2}{x + 3} - 1 = \frac{3}{x + 2},
\]

let's first simplify the left side:

1. Combine the terms on the left:

\[
\frac{x - 2}{x + 3} - 1 = \frac{x - 2 - (x + 3)}{x + 3} = \frac{x - 2 - x - 3}{x + 3} = \frac{-5}{x + 3}.
\]

Now the equation looks like this:

\[
\frac{-5}{x + 3} = \frac{3}{x + 2}.
\]

2. Cross-multiply to eliminate the fractions:

\[
-5(x + 2) = 3(x + 3).
\]

3. Distribute both sides:

\[
-5x - 10 = 3x + 9.
\]

4. Move all terms involving \(x\) to one side and constant terms to the other:

\[
-5x - 3x = 9 + 10,
\]

which simplifies to:

\[
-8x = 19.
\]

5. Solve for \(x\):

\[
x = -\frac{19}{8}.
\]

### Final Answer
The solution to the equation is 

\[
x = -\frac{19}{8}.
\]

Would you like more details on any step? Here are some related questions:

1. How do you solve rational equations in general?
2. What methods can be used to check the solution for rational equations?
3. Can you explain the concept of excluded values in rational equations?
4. How would you graph the functions involved in this equation?
5. What is the significance of cross-multiplication in solving equations?

**Tip:** Always check for any restrictions on \(x\) that could make the denominator zero when solving rational equations.

Solve for the rational equation x-2 over x +3 separate then - 1=3 over x +2

Discover how to solve the rational equation (x-2)/(x+3) - 1 = 3/(x+2) with a comprehensive step-by-step guide. This tutorial emphasizes key algebraic principles and techniques used in solving rational equations, making it ideal for students in Grades 8-10.

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