Determine the domain of the variable in the rational equation.

To determine the domain of the variable \( x \) in the given rational equation:

\[
\frac{4}{x + 2} = \frac{7}{x + 4}
\]

we need to identify any values of \( x \) that make the denominators equal to zero. Rational expressions are undefined when their denominators are zero.

### Step 1: Set each denominator equal to zero
1. For the denominator \( x + 2 \), set it equal to zero:
   \[
   x + 2 = 0
   \]
   Solving this gives:
   \[
   x = -2
   \]
   So, \( x = -2 \) is not allowed in the domain.

2. For the denominator \( x + 4 \), set it equal to zero:
   \[
   x + 4 = 0
   \]
   Solving this gives:
   \[
   x = -4
   \]
   So, \( x = -4 \) is also not allowed in the domain.

### Step 2: State the domain
The domain of the variable \( x \) includes all real numbers **except** \( x = -2 \) and \( x = -4 \), because these values make the denominators zero.

### Final Answer:
The domain of \( x \) is:
\[
x \in \mathbb{R}, \quad x \neq -2, \quad x \neq -4
\]

Would you like more details on domain restrictions or other related concepts?

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Here are some related questions you might consider:
1. What is the domain of a rational function with more complex denominators?
2. How do we find the domain of a function involving square roots in addition to rational expressions?
3. Can domain restrictions lead to extraneous solutions in rational equations?
4. How would the domain change if the equation involved products of rational expressions?
5. What is the relationship between domain restrictions and asymptotes in rational functions?

**Tip:** Always check for undefined values in any equation, particularly in rational expressions, by solving for when the denominator equals zero.

Learn how to determine the domain of the variable in a rational equation with a detailed step-by-step solution. This problem covers key concepts in algebra, including rational equations and domain restrictions. Suitable for students in Grades 9-12.

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