To solve the equation 

\[
\frac{x+2}{x^2 + 7x} = \frac{1}{x+3},
\]

we can follow these steps:

### Step 1: Identify common denominators and cross-multiply

The common denominators for the left-hand side (LHS) and the right-hand side (RHS) are already apparent in their current forms. We can proceed by cross-multiplying to eliminate the fractions:

\[
(x+2)(x+3) = 1 \cdot (x^2 + 7x).
\]

### Step 2: Expand both sides

- **Left-hand side (LHS)**: Expand \((x+2)(x+3)\)

  \[
  (x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.
  \]

- **Right-hand side (RHS)**: Keep it as \(x^2 + 7x\).

### Step 3: Set the expanded forms equal to each other

\[
x^2 + 5x + 6 = x^2 + 7x.
\]

### Step 4: Simplify the equation

Subtract \(x^2\) from both sides:

\[
5x + 6 = 7x.
\]

Now, isolate \(x\) by subtracting \(5x\) from both sides:

\[
6 = 2x.
\]

### Step 5: Solve for \(x\)

\[
x = \frac{6}{2} = 3.
\]

### Step 6: Check the solution

To ensure our solution \(x = 3\) is valid, substitute \(x = 3\) back into the original equation:

- **Original equation**: \(\frac{x+2}{x^2 + 7x} = \frac{1}{x+3}\).
  
- Substituting \(x = 3\):

  - LHS: \(\frac{3+2}{3^2 + 7 \cdot 3} = \frac{5}{9 + 21} = \frac{5}{30} = \frac{1}{6}\).
  
  - RHS: \(\frac{1}{3+3} = \frac{1}{6}\).

Since both sides are equal when \(x = 3\), this solution is valid.

### Final Answer

\[
x = 3
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What would happen if we found more than one solution for \(x\)?
2. How would the equation change if the right-hand side was a different fraction, like \(\frac{2}{x+3}\)?
3. How can we handle equations that have variables in both the numerator and the denominator?
4. What does it mean if one of the solutions makes the denominator zero?
5. How would we solve this equation if it were in a different form, like \(\frac{x+2}{x^2+7x} = \frac{2}{x+3}\)?

**Tip:** When solving equations with variables in the denominator, always check for potential restrictions on the variable that might make the denominator zero. These values need to be excluded from the solution set.

Learn how to solve the equation (x+2)/(x²+7x) = 1/(x+3) with detailed steps and explanations. Understand the concept of cross-multiplication and how to simplify fractions in algebraic equations. Suitable for high school students.

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