To solve the inequality $2|x + 8| - 5 > 3$ algebraically, follow these steps:

Step 1: Isolate the absolute value expression

Start by isolating the absolute value expression on one side of the inequality.

$2|x + 8| - 5 > 3$

Add 5 to both sides:

$2|x + 8| > 8$

Next, divide both sides by 2:

$|x + 8| > 4$

Step 2: Break the absolute value inequality into two cases

The inequality $|x + 8| > 4$ means that the expression inside the absolute value can be either greater than 4 or less than -4. So, we split this into two cases:

1. Case 1: $x + 8 > 4$
2. Case 2: $x + 8 < -4$

Step 3: Solve each case

Case 1: $x + 8 > 4$

Subtract 8 from both sides:

$x > -4$

Case 2: $x + 8 < -4$

Subtract 8 from both sides:

$x < -12$

Step 4: Combine the solutions

The solutions from the two cases give us the solution set:

$x > -4 \quad \text{or} \quad x < -12$

Step 5: Write the solution in interval notation

The solution set is the union of the two intervals:

$x \in (-\infty, -12) \cup (-4, \infty)$

This is the algebraic solution to the inequality $2|x + 8| - 5 > 3$.

Would you like more details or have any questions?

Here are five related questions for practice:

1. Solve the inequality $3|x - 5| + 2 > 11$.
2. Solve the equation $4|x + 1| = 12$.
3. What is the solution to $|2x - 3| \leq 7$?
4. Solve the compound inequality $|x + 6| < 2$ and $|x + 6| > -2$.
5. Find the solution set for $|3x + 9| \geq 15$.

Tip: When solving absolute value inequalities, always consider both the positive and negative scenarios for the expression inside the absolute value.