https://file.aiquickdraw.com/mathbot/file/1727652269510b6q3kgkx.jpg

The inequality you have uploaded is:

\[
|6x + 8| \leq 25 \quad \text{or} \quad |6y + 8| \leq 25
\]

This represents two inequalities involving absolute values. We can break them down and solve them as follows:

### 1. Solving \( |6x + 8| \leq 25 \):
By the definition of absolute value:

\[
-25 \leq 6x + 8 \leq 25
\]

Subtract 8 from all parts:

\[
-33 \leq 6x \leq 17
\]

Now divide through by 6:

\[
-5.5 \leq x \leq \frac{17}{6} \approx 2.83
\]

So, the solution for \( x \) is:

\[
x \in [-5.5, 2.83]
\]

### 2. Solving \( |6y + 8| \leq 25 \):
Similarly,

\[
-25 \leq 6y + 8 \leq 25
\]

Subtract 8 from all parts:

\[
-33 \leq 6y \leq 17
\]

Divide through by 6:

\[
-5.5 \leq y \leq \frac{17}{6} \approx 2.83
\]

So, the solution for \( y \) is:

\[
y \in [-5.5, 2.83]
\]

### 3. Combined Solution (Graphing the Inequality):

This inequality involves "or," meaning the solution is valid if either of the inequalities is true. Geometrically, this represents two strips on the coordinate plane:

- One strip where \( x \) is between \( -5.5 \) and \( 2.83 \), regardless of \( y \).
- Another strip where \( y \) is between \( -5.5 \) and \( 2.83 \), regardless of \( x \).

These two strips together form a cross-like shape on the coordinate plane. To graph this:

- Draw vertical lines at \( x = -5.5 \) and \( x = 2.83 \), and shade the region in between.
- Draw horizontal lines at \( y = -5.5 \) and \( y = 2.83 \), and shade the region in between.

The final graph will consist of a rectangle, with boundaries:
\[
-5.5 \leq x \leq 2.83 \quad \text{or} \quad -5.5 \leq y \leq 2.83
\]

Would you like a graph of this solution? Let me know if you have questions!

### Relative questions:
1. How do you solve inequalities involving absolute values?
2. What is the difference between solving inequalities with "and" vs "or"?
3. How does the graph of an absolute value inequality differ from a linear inequality?
4. How can the solution region change if the inequalities were strict (i.e., \( < \) instead of \( \leq \))?
5. How would the solution change if the coefficients inside the absolute value were different?

**Tip:** Always remember to split an absolute value inequality into two cases—positive and negative—and solve each case separately.

Graph the solution set of the inequality: |6x + 8| ≤ 25 or |6y + 8| ≤ 25.

Learn how to graph the solution set of the absolute value inequalities |6x + 8| ≤ 25 or |6y + 8| ≤ 25. This tutorial includes step-by-step instructions for solving and graphing on the coordinate plane. Suitable for high school students studying algebra.

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