We are given the inequality:

$5\left|4 + 2x\right| - 5 > 25$

Step 1: Isolate the absolute value expression.

First, add 5 to both sides of the inequality to eliminate the constant term on the left:

$5\left|4 + 2x\right| > 30$

Step 2: Divide by 5 to simplify.

Next, divide both sides of the inequality by 5:

$\left|4 + 2x\right| > 6$

Step 3: Solve the absolute value inequality.

The absolute value inequality $\left|4 + 2x\right| > 6$ can be split into two cases:

Case 1: $4 + 2x > 6$

Solve for $x$: $4 + 2x > 6$ $2x > 2$ $x > 1$

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

Solve for $x$: $4 + 2x < -6$ $2x < -10$ $x < -5$

Step 4: Combine the solutions.

The solution to the inequality is: $x < -5 \quad \text{or} \quad x > 1$

Final Answer:

The solution to the inequality is $x < -5$ or $x > 1$.

Would you like more details, or do you have any questions?

Here are some related questions:

1. What happens when absolute value inequalities have $\leq$ or $\geq$?
2. How would you solve a system of inequalities?
3. Can you solve this inequality graphically?
4. How does dividing or multiplying by negative numbers affect inequalities?
5. What is the geometric interpretation of absolute values?

Tip: Always remember that dividing or multiplying by negative numbers in inequalities reverses the inequality sign!