https://file.aiquickdraw.com/mathbot/file/1726502073652zzyaezui.jpg

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.

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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.

Доведіть нерівності, якщо x — довільне дійсне число: 1) x^2 + 4 ≥ 4x; 2) 9x^2 ≥ 12x - 4.

Learn how to prove quadratic inequalities, including x^2 + 4 ≥ 4x and 9x^2 ≥ 12x - 4, using factoring and the method of differences. This problem highlights key algebraic techniques and is suitable for students in Grades 8-10.

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