https://file.aiquickdraw.com/mathbot/file/17276535102130vgph9l8.jpg

graph the solution then say which of the 8 graphs is a match

Let's start by analyzing the given inequality:

\[
|2x + 1| \leq 12 \text{ or } |2y + 1| \leq 12
\]

This inequality consists of two absolute value inequalities combined with an "or". We will solve both parts separately and then graph their solutions.

### 1. Solving \( |2x + 1| \leq 12 \):

For the absolute value inequality \( |2x + 1| \leq 12 \), we break it into two inequalities:

\[
-12 \leq 2x + 1 \leq 12
\]

Subtract 1 from all parts:

\[
-13 \leq 2x \leq 11
\]

Now, divide by 2:

\[
-6.5 \leq x \leq 5.5
\]

This means that \( x \) is in the interval \( [-6.5, 5.5] \).

### 2. Solving \( |2y + 1| \leq 12 \):

Similarly, for the inequality \( |2y + 1| \leq 12 \):

\[
-12 \leq 2y + 1 \leq 12
\]

Subtract 1 from all parts:

\[
-13 \leq 2y \leq 11
\]

Divide by 2:

\[
-6.5 \leq y \leq 5.5
\]

This means that \( y \) is in the interval \( [-6.5, 5.5] \).

### 3. Graphing the Inequality

The inequality \( |2x + 1| \leq 12 \) corresponds to the vertical strip between \( x = -6.5 \) and \( x = 5.5 \), and the inequality \( |2y + 1| \leq 12 \) corresponds to the horizontal strip between \( y = -6.5 \) and \( y = 5.5 \).

Since these inequalities are combined with an "or", the solution will cover both the horizontal and vertical strips. This will look like two rectangles, one along the x-axis and the other along the y-axis, but combined will cover a cross-like shape.

### Matching the Graph:

Looking at the images you provided, the matching graph is **the first one in the first row** (top-left). This graph shows the shaded region that corresponds to the intervals \( [-6.5, 5.5] \) on both the x and y axes, forming a cross-like shape with both vertical and horizontal stripes shaded.

---

Would you like more details on how the graph was derived, or do you have any other questions?

Here are 5 related questions:
1. How would the graph change if the inequality were combined with an "and" instead of "or"?
2. What happens if we change the constants in the inequality (e.g., make them larger or smaller)?
3. Can we represent this inequality system in terms of two linear inequalities without the absolute value?
4. How does solving absolute value inequalities compare with solving regular linear inequalities?
5. What is the geometric meaning of combining inequalities using "or"?

**Tip:** When graphing absolute value inequalities, remember that they always create boundary lines, and you can often translate these boundaries into simpler linear inequalities for easier graphing.

Graph the solution set of the following linear inequality: |2x + 1| ≤ 12 or |2y + 1| ≤ 12.

Graph the solution set of the absolute value inequalities |2x + 1| ≤ 12 or |2y + 1| ≤ 12. This tutorial includes step-by-step methods for solving and graphing compound inequalities and is suitable for high school students.

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