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The solution to the inequality \(\frac{1}{x+1} + \frac{1}{x} < \frac{1}{x(x+1)}\) is:

\[
x \in (-\infty, -1)
\]

This means that \(x\) must be less than \(-1\) for the inequality to hold.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What are the steps to solve rational inequalities like this one?
2. How would the inequality change if the right-hand side were greater than or equal to the left-hand side?
3. What happens to the inequality if \(x = -1\)?
4. How can we solve rational inequalities with higher degrees in the denominator?
5. What are the restrictions or exclusions for the domain of this inequality?

**Tip:** When solving inequalities involving rational expressions, always check for values that make the denominators zero, as these points are excluded from the solution.

Solve the inequality \(\frac{1}{x+1} + \frac{1}{x} < \frac{1}{x(x+1)}\)

This guide provides a step-by-step solution to the inequality \(\frac{1}{x+1} + \frac{1}{x} < \frac{1}{x(x+1)}\), covering algebraic concepts, rational expressions, and properties of inequalities. Suitable for Grades 10-12 students.

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