To solve the equation \( 9 + 2|4x - 5| = 13 \), let's proceed step by step.

### Step 1: Isolate the absolute value expression.
\[
9 + 2|4x - 5| = 13
\]
Subtract 9 from both sides:
\[
2|4x - 5| = 13 - 9
\]
\[
2|4x - 5| = 4
\]

### Step 2: Solve for the absolute value term.
Divide both sides by 2:
\[
|4x - 5| = 2
\]

### Step 3: Break it into two cases since the absolute value represents two possible equations.

#### Case 1: \( 4x - 5 = 2 \)
Solve for \( x \):
\[
4x = 2 + 5
\]
\[
4x = 7
\]
\[
x = \frac{7}{4}
\]

#### Case 2: \( 4x - 5 = -2 \)
Solve for \( x \):
\[
4x = -2 + 5
\]
\[
4x = 3
\]
\[
x = \frac{3}{4}
\]

### Final Answer:
The solutions are:
\[
x = \frac{7}{4} \quad \text{or} \quad x = \frac{3}{4}
\]

Would you like more details on any step? Feel free to ask!

Here are some related questions you might find helpful:
1. How do we solve more complex absolute value equations?
2. What happens if the equation has no solutions?
3. How can we graph absolute value functions?
4. What is the significance of absolute values in real-world applications?
5. How do we handle inequalities with absolute values?

**Tip**: Always remember to break down absolute value equations into two cases: one for the positive and one for the negative scenario.

Learn how to solve the absolute value equation 9 + 2|4x − 5| = 13 with a step-by-step solution. This problem explores key algebraic concepts like isolating absolute values and breaking them into two cases. Suitable for students in Grades 8-10.

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