We are given the equation:

\[
4 \mid 2y - 7 \mid + 13 = 9
\]

### Step 1: Isolate the absolute value term.

Subtract 13 from both sides:

\[
4 \mid 2y - 7 \mid = 9 - 13
\]
\[
4 \mid 2y - 7 \mid = -4
\]

### Step 2: Analyze the result.

The equation now reads:

\[
4 \mid 2y - 7 \mid = -4
\]

This is impossible because the absolute value of any expression is always non-negative, and multiplying it by a positive number (in this case, 4) will not result in a negative value like \(-4\).

### Conclusion:
There is **no solution** to this equation.

Would you like further clarification on this? Here are some related questions to explore:

1. How do absolute value equations work?
2. What are the properties of absolute values?
3. How do we handle inequalities with absolute values?
4. Can absolute value functions have negative results?
5. How would the equation change if the right-hand side were positive?

**Tip:** Always check the range of possible values when dealing with absolute values!

Explore the solution to the absolute value equation 4|2y − 7| + 13 = 9. This step-by-step guide covers absolute value properties and algebraic manipulation. Learn why some absolute value equations may have no solution and understand the conditions that make these equations solvable. Ideal for students in Grades 9-11.

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