The given equation is:

$1 + 3|2x + 1| = 16$

Step 1: Isolate the absolute value term

First, subtract 1 from both sides of the equation to isolate the absolute value expression:

$3|2x + 1| = 15$

Step 2: Solve for the absolute value

Now, divide both sides by 3 to fully isolate the absolute value:

$|2x + 1| = 5$

Step 3: Set up two equations

The absolute value equation $|2x + 1| = 5$ splits into two possible cases:

1. $2x + 1 = 5$
2. $2x + 1 = -5$

Step 4: Solve both cases

Case 1: $2x + 1 = 5$

Subtract 1 from both sides:

$2x = 4$

Now, divide by 2:

$x = 2$

Case 2: $2x + 1 = -5$

Subtract 1 from both sides:

$2x = -6$

Now, divide by 2:

$x = -3$

Step 5: Final solutions

The solutions are:

$x = 2 \quad \text{or} \quad x = -3$

Would you like further details or have any questions about this solution? Here are 5 related questions to expand on this:

1. How do we solve more complex absolute value equations?
2. What happens if there are two absolute value terms in an equation?
3. How do we graph absolute value functions?
4. What real-world problems use absolute value equations?
5. How do absolute value inequalities differ from equations?

Tip: Always remember, the absolute value equation $|A| = B$ has two cases: $A = B$ and $A = -B$.