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We are tasked with graphing the solution set of the following linear inequality:

\[
|8x + 5| \leq 29 \quad \text{or} \quad |8y + 5| \leq 29
\]

### Step-by-Step Solution:

#### 1. Solve for \( x \) in \( |8x + 5| \leq 29 \):
The absolute value inequality can be split into two parts:
\[
-29 \leq 8x + 5 \leq 29
\]
- Subtract 5 from all parts:
\[
-34 \leq 8x \leq 24
\]
- Divide by 8:
\[
-\frac{34}{8} \leq x \leq \frac{24}{8}
\]
Simplifying the fractions:
\[
-4.25 \leq x \leq 3
\]
This gives the interval \( x \in [-4.25, 3] \).

#### 2. Solve for \( y \) in \( |8y + 5| \leq 29 \):
Following the same steps:
\[
-29 \leq 8y + 5 \leq 29
\]
- Subtract 5 from all parts:
\[
-34 \leq 8y \leq 24
\]
- Divide by 8:
\[
-\frac{34}{8} \leq y \leq \frac{24}{8}
\]
This simplifies to:
\[
-4.25 \leq y \leq 3
\]
Thus, \( y \in [-4.25, 3] \).

#### 3. Graphing the Solution Set:
- The solution set is a rectangular region on the coordinate plane, where:
  - \( x \in [-4.25, 3] \)
  - \( y \in [-4.25, 3] \)
- This means the graph represents a rectangle with the vertices at:
  \[
  (-4.25, -4.25), \quad (-4.25, 3), \quad (3, -4.25), \quad (3, 3)
  \]
- The graph would show a solid line around this rectangle, as the inequalities are inclusive (using \( \leq \)).

Would you like a detailed explanation of the graph or further clarification?

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Here are 5 related questions to expand this topic:
1. How do we handle systems of inequalities when graphing?
2. What is the difference between strict and non-strict inequalities in graphs?
3. How do we graph a system of inequalities involving circles (absolute values with two variables)?
4. How would the graph change if the inequality was \( |8x + 5| < 29 \)?
5. What are the key differences between solving absolute value equations and inequalities?

**Tip:** When solving absolute value inequalities, always remember to split the inequality into two cases: positive and negative.

Graph the solution set of the following linear inequality: |8x + 5| ≤ 29 or |8y + 5| ≤ 29.

Learn how to graph the solution set for the inequality |8x + 5| ≤ 29 or |8y + 5| ≤ 29. The solution involves solving absolute value inequalities and plotting the solution region on a coordinate plane. Suitable for students in Grades 9-11.

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