$$ {\displaylines\left\vert\frac{y+9}{4}\right\vert=5\\ } $$

We are tasked with solving the equation:

\[
\left\vert \frac{y+9}{4} \right\vert = 5
\]

To solve an absolute value equation, we consider both the positive and negative cases. So, we will break this into two separate equations:

### Case 1: Positive
\[
\frac{y+9}{4} = 5
\]
Multiply both sides by 4 to eliminate the denominator:
\[
y + 9 = 20
\]
Subtract 9 from both sides:
\[
y = 11
\]

### Case 2: Negative
\[
\frac{y+9}{4} = -5
\]
Multiply both sides by 4:
\[
y + 9 = -20
\]
Subtract 9 from both sides:
\[
y = -29
\]

Thus, the solutions to the equation are:
\[
y = 11 \quad \text{or} \quad y = -29
\]

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How do you solve absolute value inequalities?
2. What happens if there's no solution to an absolute value equation?
3. Can absolute value equations have only one solution?
4. How does the absolute value function affect the graph of an equation?
5. What are the real-world applications of absolute value equations?

**Tip:** When solving absolute value equations, always check both the positive and negative cases to capture all possible solutions.

Solve the absolute value equation: |(y + 9)/4| = 5

Learn how to solve the absolute value equation |(y + 9)/4| = 5 with step-by-step explanations. This solution breaks down both the positive and negative cases of the equation, showing how to find two possible solutions. Ideal for students in Grades 8-10 learning about absolute value and linear equations.

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