https://file.aiquickdraw.com/mathbot/file/17266242636419pqo324n.jpg

The inequality to solve is:

\[
|2x + 7| \leq 19
\]

### Step-by-Step Solution:

To solve the absolute value inequality \( |2x + 7| \leq 19 \), we split it into two inequalities:

1. \( 2x + 7 \leq 19 \)
2. \( -(2x + 7) \leq 19 \)

#### Solve the first inequality:
\[
2x + 7 \leq 19
\]
Subtract 7 from both sides:
\[
2x \leq 12
\]
Divide by 2:
\[
x \leq 6
\]

#### Solve the second inequality:
\[
-(2x + 7) \leq 19
\]
Distribute the negative:
\[
-2x - 7 \leq 19
\]
Add 7 to both sides:
\[
-2x \leq 26
\]
Divide by -2 (flip the inequality sign):
\[
x \geq -13
\]

### Solution Set:

The solution is the intersection of \( x \leq 6 \) and \( x \geq -13 \), which gives the interval:

\[
x \in [-13, 6]
\]

In interval notation, the solution set is:

\[
[-13, 6]
\]

### Graph:

The graph would show a closed interval from -13 to 6, with filled circles on both -13 and 6 to indicate that these values are included in the solution set.

Would you like further details or a deeper explanation?

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Here are 5 related questions:
1. How do you solve absolute value equations that involve strict inequalities like \( |2x + 7| < 19 \)?
2. What are the steps to graph an inequality on a number line?
3. How do the rules change if the inequality sign is \( \geq \) or \( \leq \)?
4. What happens if there is no intersection between the two parts of the inequality?
5. How can you express the union of solution sets in interval notation?

**Tip:** Always remember to flip the inequality sign when dividing or multiplying by a negative number in inequalities.

Learn how to solve the absolute value inequality |2x + 7| ≤ 19 step by step. This solution covers breaking the inequality into two cases and solving for x in each case, suitable for students in Grades 7-10.

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