We are given the equation:

\[
| -2b + 11 | = 17
\]

To solve this, we need to consider two cases, as the absolute value expression can be either positive or negative.

### Case 1: \( -2b + 11 = 17 \)

Solve for \( b \):

\[
-2b + 11 = 17
\]

Subtract 11 from both sides:

\[
-2b = 17 - 11
\]

\[
-2b = 6
\]

Now, divide both sides by -2:

\[
b = \frac{6}{-2} = -3
\]

### Case 2: \( -2b + 11 = -17 \)

Solve for \( b \):

\[
-2b + 11 = -17
\]

Subtract 11 from both sides:

\[
-2b = -17 - 11
\]

\[
-2b = -28
\]

Now, divide both sides by -2:

\[
b = \frac{-28}{-2} = 14
\]

### Final Answer:
The solutions are \( b = -3 \) and \( b = 14 \).

Would you like a further explanation or clarification? Here are 5 related questions to practice:

1. Solve \( |3x - 7| = 10 \).
2. Find \( |4y + 5| = 13 \).
3. If \( |x - 2| = 5 \), what are the values of \( x \)?
4. Solve \( |5k - 9| = 8 \).
5. Solve \( |p + 3| = 0 \).

**Tip:** When solving absolute value equations, always remember to account for both the positive and negative possibilities.

Learn how to solve the absolute value equation | -2b + 11 | = 17 using two cases, considering both positive and negative solutions. This problem explores key algebraic concepts such as solving linear equations and applying the absolute value theorem. Suitable for students in Grades 7-10.

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