Let's solve the given inequality algebraically:

The inequality is:

$5|4 + 2x| - 5 > 25$

Step-by-step solution:

1. Isolate the absolute value: Add 5 to both sides of the inequality: $5|4 + 2x| - 5 + 5 > 25 + 5$ $5|4 + 2x| > 30$

2. Divide by 5: $|4 + 2x| > 6$

3. Remove the absolute value: To solve this, we split it into two cases:

   • Case 1: $4 + 2x > 6$
   • Case 2: $4 + 2x < -6$

   Case 1: $4 + 2x > 6$

   Subtract 4 from both sides: $2x > 2$ Divide by 2: $x > 1$

   Case 2: $4 + 2x < -6$

   Subtract 4 from both sides: $2x < -10$ Divide by 2: $x < -5$

Final solution:

The solution to the inequality is:

$x > 1 \quad \text{or} \quad x < -5$

In interval notation, this can be written as:

$(-\infty, -5) \cup (1, \infty)$

Let me know if you would like further details or if you have any questions!

Here are 5 related questions:

1. How do you solve inequalities involving absolute values?
2. What happens when the inequality has a "less than" sign instead of "greater than" for absolute values?
3. Can you explain the graphical interpretation of absolute value inequalities?
4. How do you handle cases where the inequality has no solution?
5. What are the different methods to solve inequalities?

Tip: Always check your solutions for absolute value inequalities by substituting values from each interval back into the original inequality.