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The problem asks to find the value of \( a \), given that \( b \) is positive and the inequality is:

\[
\frac{b + 1}{a} < \frac{1}{b}
\]

### Solution:

1. **Start with the given inequality:**
   \[
   \frac{b + 1}{a} < \frac{1}{b}
   \]

2. **Multiply both sides of the inequality by \( ab \) to eliminate the denominators** (since \( b > 0 \), the direction of the inequality remains the same):
   \[
   b(b + 1) < a
   \]

3. **Simplify the left-hand side:**
   \[
   b^2 + b < a
   \]

Thus, the inequality becomes:
\[
a > b^2 + b
\]

### Conclusion:

For \( a \) to satisfy the inequality, it must be **greater than \( b^2 + b \)**. Since \( b \) is positive, the solution for \( a \) is any number larger than \( b^2 + b \).

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Do you want more details on this? Here are 5 related questions to consider:

1. What happens if \( b \) is negative or zero in this inequality?
2. How would the solution change if the inequality were \( \frac{b + 1}{a} \leq \frac{1}{b} \)?
3. Can you express \( a \) as a function of \( b \) in this inequality?
4. What are the real-world applications of inequalities involving fractions?
5. How does this inequality change if additional constraints are placed on \( a \)?

**Tip:** When solving inequalities, multiplying or dividing by a variable requires extra caution, especially with positive or negative values, as it can reverse the inequality!

Find the value of a, given that b is positive and the inequality \(\frac{b + 1}{a} < \frac{1}{b}\).

Learn how to solve an inequality where \( b \) is positive, and find the conditions for \( a \). This example shows step-by-step how to manipulate an inequality involving fractions to find that \( a > b^2 + b \). Suitable for students in Grades 8-10.

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