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!