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We are given the compound inequality:

\[
2x - 4 \geq -16 \quad \text{and} \quad 4x + 5 > 25
\]

### Step 1: Solve the first inequality \( 2x - 4 \geq -16 \)

1. Add 4 to both sides:
   \[
   2x \geq -16 + 4
   \]
   \[
   2x \geq -12
   \]

2. Divide both sides by 2:
   \[
   x \geq -6
   \]

### Step 2: Solve the second inequality \( 4x + 5 > 25 \)

1. Subtract 5 from both sides:
   \[
   4x > 25 - 5
   \]
   \[
   4x > 20
   \]

2. Divide both sides by 4:
   \[
   x > 5
   \]

### Step 3: Find the intersection

We need to satisfy both inequalities:
1. \( x \geq -6 \)
2. \( x > 5 \)

Since \( x > 5 \) is stricter than \( x \geq -6 \), the solution to the compound inequality is \( x > 5 \).

### Final Answer (in interval notation):

The solution in interval notation is:
\[
(5, \infty)
\]

Would you like more details or have any questions?

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Here are 5 related questions to further your understanding:
1. How would the solution change if the first inequality were strict (i.e., \( 2x - 4 > -16 \))?
2. What happens to the solution if one of the inequalities were reversed?
3. How do you graph the solution \( x > 5 \) on a number line?
4. What would the compound inequality look like if both inequalities had no overlap?
5. How can inequalities be combined with union instead of intersection?

**Tip:** Always ensure that you interpret the relationship between the inequalities (i.e., conjunction or disjunction) when solving compound inequalities.

Solve the compound inequality: 2x - 4 ≥ -16 and 4x + 5 > 25. Write the solution in interval notation.

Learn how to solve compound inequalities by solving 2x - 4 ≥ -16 and 4x + 5 > 25. This problem involves key algebraic techniques like solving linear inequalities and finding intersections, suitable for students in Grades 9-10.

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