https://file.aiquickdraw.com/mathbot/file/17271463155444pldo9gj.jpg

The equation to solve is:

\[
|2b - 9| = |b - 6|
\]

To solve this absolute value equation, we need to consider the different cases where the expressions inside the absolute value can be positive or negative.

### Case 1: Both expressions are positive or equal
In this case, we assume:
\[
2b - 9 = b - 6
\]

Solving this:

\[
2b - b = -6 + 9
\]
\[
b = 3
\]

### Case 2: One expression is negative, the other positive
For the next two cases, we handle the situation where one of the expressions inside the absolute value is negative.

#### Subcase 2a: \( 2b - 9 = -(b - 6) \)
\[
2b - 9 = -b + 6
\]

Solving this:

\[
2b + b = 6 + 9
\]
\[
3b = 15
\]
\[
b = 5
\]

#### Subcase 2b: \( -(2b - 9) = b - 6 \)
\[
-2b + 9 = b - 6
\]

Solving this:

\[
-2b - b = -6 - 9
\]
\[
-3b = -15
\]
\[
b = 5
\]

### Conclusion
The solutions for \( b \) are:
\[
b = 3 \quad \text{and} \quad b = 5
\]

Let me know if you'd like further explanation on any of the steps.

Here are 5 related questions you might find useful:
1. How do you solve absolute value equations in general?
2. What happens when the absolute values in an equation are on different sides?
3. Can an absolute value equation have no solution? If yes, when?
4. How do you solve absolute value inequalities?
5. What is the geometric interpretation of absolute value equations?

**Tip**: Always check your solutions by plugging them back into the original absolute value equation to ensure both satisfy the equation.

Learn how to solve the absolute value equation |2b - 9| = |b - 6| with a detailed step-by-step solution. This problem covers key concepts of absolute value and algebraic manipulation, suitable for students in Grades 8-10.

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