https://file.aiquickdraw.com/mathbot/file/1728816092536l4d3oh49.jpg

The equation in the image is:

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

To solve for \(x\), we will follow these steps:

### Step 1: Cross-multiply
Using the cross-multiplication method, multiply both sides of the equation:

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

This simplifies to:

\[
3x = 2(x + 1)
\]

### Step 2: Distribute the 2 on the right-hand side
Distribute the 2 to both terms inside the parentheses:

\[
3x = 2x + 2
\]

### Step 3: Move terms involving \(x\) to one side
Subtract \(2x\) from both sides:

\[
3x - 2x = 2
\]

Simplify:

\[
x = 2
\]

Thus, the solution is:

\[
x = 2
\]

Would you like more details or explanations? Here are some follow-up questions to further explore this topic:

1. How do you verify the solution by substituting \(x = 2\) back into the original equation?
2. Can you explain why cross-multiplying works in this type of equation?
3. How does the distributive property help in simplifying algebraic equations?
4. What would happen if the denominator had been a different expression, such as \(x + 2\)?
5. Can you solve a similar equation with different fractions on both sides?

**Tip:** Always check for restrictions on variables in rational equations, such as when denominators could be zero.

Solve the equation \(\frac{x}{x + 1} = \frac{2}{3}\).

Learn how to solve the rational equation \(\frac{x}{x + 1} = \frac{2}{3}\) by cross-multiplication. This step-by-step solution involves algebraic manipulation, distribution, and isolating the variable. Suitable for students in Grades 7-9.

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