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?

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.