To solve the given inequality, let's first carefully review the expression from the original image:

$x + y = x^{y+1} - x^{y-1}$

It appears you're now asking for solving an inequality. However, no inequality was initially provided.

Let me assume a possible inequality based on the provided equation. If we were to consider the equation as an inequality, for example:

$x + y < x^{y+1} - x^{y-1}$

We need to find the values of $x$ and $y$ that satisfy this inequality.

Solving the Inequality

1. Rewrite the inequality:

   $x + y < x^{y+1} - x^{y-1}$

2. Simplify and rearrange the inequality:

   To approach this problem, let’s examine specific cases for $x$ and $y$ to make it more manageable. For example, if $x > 0$ and $y > 0$, the inequality could behave differently compared to other combinations of signs for $x$ and $y$.

   Let's simplify:

   $x + y < x \cdot x^y - \frac{x^y}{x}$

   This can be rewritten as:

   $x + y < x^{y+1} - x^{y-1}$

   Rearrange to focus on terms:

   $x + y - x^{y+1} + x^{y-1} < 0$

3. Analyze the inequality:

   The solution depends on understanding how the exponential terms compare with the linear terms. Typically, there’s no straightforward algebraic simplification. We would analyze it by examining values or graphical methods.

Determining Solution Intervals

Given the complexity, solving such an equation analytically isn’t straightforward. Instead, we would test specific values or ranges to determine intervals.

For simplicity, if we were looking for $x > 1$ and $y > 1$, the expression:

$x^{y+1} - x^{y-1} \text{ grows much faster than } x + y$

Indicating that typically, for larger positive $x, y$, the inequality might hold. However, with variables involved and no bounds, numerical methods or graphing tools are necessary.

Concluding the Solution

To accurately answer the question based on interval notation:

1. Check possible cases: Test various intervals for $x$ and $y$ (positive or negative) to see