We are given the inequality:

\[
|2x + 1| \leq 19
\]

To solve this, we'll break it down into two cases based on the definition of absolute value.

### Step 1: Break into two inequalities
By the definition of absolute value:

\[
-19 \leq 2x + 1 \leq 19
\]

### Step 2: Solve each inequality

#### Solve \( -19 \leq 2x + 1 \):

1. Subtract 1 from both sides:
   \[
   -19 - 1 \leq 2x
   \]
   \[
   -20 \leq 2x
   \]

2. Divide by 2:
   \[
   -10 \leq x
   \]
   
#### Solve \( 2x + 1 \leq 19 \):

1. Subtract 1 from both sides:
   \[
   2x \leq 19 - 1
   \]
   \[
   2x \leq 18
   \]

2. Divide by 2:
   \[
   x \leq 9
   \]

### Step 3: Combine the results
From the two inequalities, we have:

\[
-10 \leq x \leq 9
\]

Thus, the solution is:

\[
x \in [-10, 9]
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you solve strict inequalities like \(|2x + 1| < 19\)?
2. How would the solution change if the inequality was \(|2x + 1| \geq 19\)?
3. Can you explain how to graph the solution on a number line?
4. What if we have \(|2x + 1| = 19\)? How would we solve that?
5. How would the steps change if the absolute value inequality involved a quadratic term?

**Tip**: Always remember to break absolute value inequalities into two cases: one positive and one negative!

Learn how to solve the absolute value inequality |2x + 1| ≤ 19 with a clear step-by-step breakdown. This algebra problem teaches how to handle absolute value inequalities and involves solving linear equations. Suitable for students in Grades 8-10.

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