https://file.aiquickdraw.com/mathbot/file/1729173976714lbifgsvb.jpg

Let's analyze the equation from the image:

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

### Step 1: Eliminate the denominators.
To simplify the equation, multiply every term by \( x + 1 \) to get rid of the fractions:

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

The \( x + 1 \) terms will cancel out in the fractions:

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

### Step 2: Expand and simplify.
Expand the terms on the left-hand side:

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

Simplify:

\[
x^2 - x = 2
\]

### Step 3: Rearrange the equation.
Move all terms to one side:

\[
x^2 - x - 2 = 0
\]

### Step 4: Solve the quadratic equation.
The equation \( x^2 - x - 2 = 0 \) is a quadratic equation, which can be factored as:

\[
(x - 2)(x + 1) = 0
\]

So, the solutions are:

\[
x - 2 = 0 \quad \text{or} \quad x + 1 = 0
\]

This gives:

\[
x = 2 \quad \text{or} \quad x = -1
\]

### Step 5: Check for extraneous solutions.
Since the original equation has denominators involving \( x + 1 \), we must exclude any solution that makes the denominator zero. Notice that for \( x = -1 \), the denominator \( x + 1 \) becomes zero. Therefore, \( x = -1 \) is not a valid solution.

Thus, the only solution is:

\[
x = 2
\]

Would you like further explanation or have any questions?

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Here are 5 related questions for further exploration:
1. How do you solve quadratic equations using the quadratic formula?
2. What are some common techniques for solving rational equations?
3. How do you identify and handle extraneous solutions?
4. What are the steps involved in factoring a quadratic expression?
5. How do you graphically interpret the solution of \( x^2 - x - 2 = 0 \)?

**Tip:** Always check for extraneous solutions, especially when dealing with equations that have variables in the denominator!

Solve for all values of x in the equation: x - (2x / (x + 1)) = 2 / (x + 1).

Explore how to solve the rational equation x - (2x / (x + 1)) = 2 / (x + 1). This step-by-step guide covers the process of eliminating denominators, simplifying, and solving quadratic equations, while addressing extraneous solutions. Perfect for students in Grades 9-11.

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