The given equation is:

\[
|5b - \frac{8}{9}| = 9
\]

This is an absolute value equation, and absolute value equations typically have two possible cases. Let's solve it step by step.

### Step 1: Define the Two Cases
For any absolute value equation \(|A| = B\), the equation can be split into two cases:
1. \(A = B\)
2. \(A = -B\)

In this case:
1. \(5b - \frac{8}{9} = 9\)
2. \(5b - \frac{8}{9} = -9\)

### Step 2: Solve Each Case Separately

#### Case 1: \(5b - \frac{8}{9} = 9\)
1. Add \(\frac{8}{9}\) to both sides:
   \[
   5b = 9 + \frac{8}{9}
   \]
2. Convert 9 into a fraction: 
   \[
   9 = \frac{81}{9}
   \]
3. Add the fractions:
   \[
   5b = \frac{81}{9} + \frac{8}{9} = \frac{89}{9}
   \]
4. Divide both sides by 5:
   \[
   b = \frac{89}{9} \times \frac{1}{5} = \frac{89}{45}
   \]

#### Case 2: \(5b - \frac{8}{9} = -9\)
1. Add \(\frac{8}{9}\) to both sides:
   \[
   5b = -9 + \frac{8}{9}
   \]
2. Convert -9 into a fraction:
   \[
   -9 = \frac{-81}{9}
   \]
3. Add the fractions:
   \[
   5b = \frac{-81}{9} + \frac{8}{9} = \frac{-73}{9}
   \]
4. Divide both sides by 5:
   \[
   b = \frac{-73}{9} \times \frac{1}{5} = \frac{-73}{45}
   \]

### Step 3: Final Solutions
The solutions for \(b\) are:
\[
b = \frac{89}{45} \quad \text{or} \quad b = \frac{-73}{45}
\]

Would you like more details on any of these steps? Here are some related questions:

1. How do you handle absolute value equations in general?
2. How do you add fractions with different numerators?
3. What are other methods to solve absolute value problems?
4. How do you convert improper fractions into mixed numbers?
5. How does dividing by a fraction work?

**Tip:** When dealing with absolute value equations, always remember to check for both the positive and negative cases!

Explore how to solve the absolute value equation |5b - 8/9| = 9 using algebraic techniques. This problem involves adding fractions and handling absolute value expressions, suitable for students in Grades 8-10.

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