Let's solve the equation:

$|4x + 10| + 6 = 11$

Step 1: Isolate the absolute value expression.

Subtract 6 from both sides to isolate the absolute value term:

$|4x + 10| = 11 - 6$

$|4x + 10| = 5$

Step 2: Set up two equations.

Since $|A| = B$ implies $A = B$ or $A = -B$, we now split the equation into two cases:

1. $4x + 10 = 5$
2. $4x + 10 = -5$

Step 3: Solve each case.

Case 1: $4x + 10 = 5$

$4x = 5 - 10$

$4x = -5$

$x = -\frac{5}{4}$

Case 2: $4x + 10 = -5$

$4x = -5 - 10$

$4x = -15$

$x = -\frac{15}{4}$

Final Solution:

The solutions are:

$x = -\frac{5}{4} \quad \text{or} \quad x = -\frac{15}{4}$

Would you like more details on any step?

Here are five related questions to expand this concept:

1. How do you solve absolute value inequalities?
2. What happens when the absolute value term is negative?
3. How do absolute value equations change in higher dimensions (like vectors)?
4. How can we graph absolute value functions?
5. How do you solve a system of equations involving absolute values?

Tip: Always check your solutions by plugging them back into the original equation to ensure accuracy.