To solve the inequality 

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

let's follow these steps:

### Step 1: Combine like terms
We can rewrite the inequality as:

\[
\frac{2}{x+1} > \frac{x(x+1) - 1}{x+1}
\]

Since the right-hand side already has a common denominator \(x+1\), we can subtract the fractions:

\[
\frac{2}{x+1} - \frac{x(x+1) - 1}{x+1} > 0
\]

### Step 2: Simplify the expression
Combine the fractions:

\[
\frac{2 - \left(x(x+1) - 1\right)}{x+1} > 0
\]

Now, expand and simplify the numerator:

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

So the inequality becomes:

\[
\frac{3 - x^2 - x}{x+1} > 0
\]

### Step 3: Factor the numerator
The expression in the numerator can be factored:

\[
\frac{-(x^2 + x - 3)}{x+1} > 0
\]

We can factor the quadratic \(x^2 + x - 3\) as:

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

So, the inequality is:

\[
\frac{-(x - 1)(x + 3)}{x+1} > 0
\]

### Step 4: Analyze the sign changes
We need to determine the intervals where the expression is positive. The critical points occur where the numerator or denominator is zero:

- \(x - 1 = 0 \Rightarrow x = 1\)
- \(x + 3 = 0 \Rightarrow x = -3\)
- \(x + 1 = 0 \Rightarrow x = -1\)

We will now test the intervals determined by these points: 

1. \(x < -3\)
2. \(-3 < x < -1\)


Learn how to solve the inequality 2/(x+1) > x - 1/(x+1) with a detailed step-by-step solution. This problem involves rational functions and quadratic equations, requiring careful analysis and interval testing. Suitable for students in Grades 10-12.

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