Step-by-step explanation:
To solve a rational equation in algebra, follow these steps:
1. **Identify the domain**: Determine any values that would make the denominator(s) equal to zero. These values must be excluded from the solution set.
2. **Clear the denominators**: Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.
3. **Simplify**: Simplify both sides of the equation as much as possible.
4. **Solve for the variable**: After clearing the denominators, solve the resulting equation for the variable.
5. **Check for extraneous solutions**: Since multiplying both sides of the equation by the LCD can introduce extraneous solutions, make sure to check your solutions in the original equation to ensure they are valid.
Let's illustrate these steps with an example:
**Example:**
Solve the rational equation \( \frac{x}{x - 3} = \frac{3}{x + 2} \).
**Solution:**
1. **Identify the domain**:
- The denominator \( x - 3 \) cannot be zero, so \( x \neq 3 \).
- The denominator \( x + 2 \) cannot be zero, so \( x \neq -2 \).
2. **Clear the denominators**:
- Multiply both sides of the equation by \( (x - 3)(x + 2) \), the least common denominator.
\[ (x - 3)(x + 2) \left( \frac{x}{x - 3} \right) = (x - 3)(x + 2) \left( \frac{3}{x + 2} \right) \]
3. **Simplify**:
\[ x(x + 2) = 3(x - 3) \]
4. **Solve for the variable**:
\[ x^2 + 2x = 3x - 9 \]
\[ x^2 - x - 9 = 0 \]
5. **Solve the quadratic equation**:
- You can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
6. **Check for extraneous solutions**:
- After finding the solutions, plug them back into the original equation to ensure they are valid solutions.
That's the general process for solving rational equations in algebra. Make sure to follow each step carefully and check your solutions at the end to ensure accuracy.
