The image contains two inequalities that you are asked to prove using the method of differences. Let's work through each one step by step.

1) Proving the inequality $x^2 + 4 \geq 4x$:

Step 1: Rewrite the inequality. $x^2 + 4 \geq 4x$ Move all terms to one side: $x^2 - 4x + 4 \geq 0$

Step 2: Factor the expression. Notice that the left-hand side is a perfect square: $(x - 2)^2 \geq 0$

Step 3: Analyze the inequality. The square of any real number is always non-negative, i.e., $(x - 2)^2 \geq 0$ for all $x \in \mathbb{R}$.

Therefore, the inequality holds for all real numbers $x$, with equality when $x = 2$.

2) Proving the inequality $9x^2 \geq 12x - 4$:

Step 1: Rewrite the inequality. $9x^2 \geq 12x - 4$ Move all terms to one side: $9x^2 - 12x + 4 \geq 0$

Step 2: Factor the quadratic expression. The expression can be factored as follows: $(3x - 2)^2 \geq 0$

Step 3: Analyze the inequality. Again, the square of any real number is always non-negative, i.e., $(3x - 2)^2 \geq 0$ for all $x \in \mathbb{R}$.

Thus, this inequality holds for all real numbers $x$, with equality when $x = \frac{2}{3}$.

Conclusion:

Both inequalities hold for all real numbers, with specific points where equality is achieved.

Would you like any additional explanations or details on these solutions?

Here are 5 follow-up questions related to this topic:

1. How do you apply the method of differences to other inequalities?
2. What are the key steps in factoring quadratic expressions?
3. Why is a perfect square always non-negative?
4. How can you check for the equality points in inequalities?
5. How do these inequalities change if the constants are modified?

Tip: Always check if you can factor an expression, as this often simplifies solving inequalities.