5$$ \left\vert4+2x\right\vert $$-5$$ >25 $$

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!

Learn how to solve the absolute value inequality 5|4 + 2x| - 5 > 25 with a detailed step-by-step solution. This problem demonstrates key algebraic concepts like absolute value inequalities and linear inequalities. Suitable for students in Grades 9-11.

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