https://file.aiquickdraw.com/mathbot/file/1725845841790nsa3nqv2.jpg

The equation in the image is:

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

Let me solve this equation step by step.

### Step 1: Find a common denominator.
The denominators are \( x \) and \( x+2 \). The least common denominator (LCD) is \( x(x+2) \).

### Step 2: Rewrite the equation with the common denominator.

\[
\frac{1}{x} + \frac{3}{x+2} = 2
\]
Multiply both sides of the equation by \( x(x+2) \) to eliminate the denominators:

\[
\left(\frac{1}{x} \cdot x(x+2)\right) + \left(\frac{3}{x+2} \cdot x(x+2)\right) = 2 \cdot x(x+2)
\]

Simplifying each term:

\[
(x+2) + 3x = 2x(x+2)
\]

### Step 3: Expand and simplify both sides.

Left side:
\[
x + 2 + 3x = 4x + 2
\]

Right side:
\[
2x(x+2) = 2x^2 + 4x
\]

So, the equation is now:

\[
4x + 2 = 2x^2 + 4x
\]

### Step 4: Move all terms to one side of the equation.

Subtract \( 4x \) from both sides:

\[
2 = 2x^2
\]

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

Divide both sides by 2:

\[
1 = x^2
\]

Take the square root of both sides:

\[
x = \pm 1
\]

### Step 6: Check for extraneous solutions.
Substitute \( x = 1 \) and \( x = -1 \) into the original equation to ensure they are valid.

1. For \( x = 1 \):

\[
\frac{1}{1} + \frac{3}{1+2} = 1 + 1 = 2 \quad \text{(valid solution)}
\]

2. For \( x = -1 \):

The denominator in \( \frac{1}{x} \) becomes zero, which is undefined. Therefore, \( x = -1 \) is not a valid solution.

### Final Solution:
\[
x = 1
\]

Would you like more details or have any questions? Here are some related questions to explore further:

1. How do you handle equations with more complex denominators?
2. Can you explain how to verify solutions for rational equations?
3. What are the common errors to avoid when solving rational equations?
4. How do you solve rational equations with multiple variables?
5. What happens if the equation has no solution or infinitely many solutions?

**Tip:** Always check for extraneous solutions, especially when solving rational equations with variables in the denominator.

Learn how to solve the rational equation 1/x + 3/(x+2) = 2 with a detailed step-by-step solution. This problem includes finding the least common denominator (LCD) and solving a quadratic equation, making it suitable for students in Grades 8-10.

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