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The image you've uploaded contains the following mathematical inequality:

Let \( x \) and \( y \) be non-negative real numbers. The inequality is given as:

\[
\frac{x}{2} \geq \sqrt{xy + y}
\]

This inequality seems to relate to certain properties of real numbers and might involve algebraic manipulation or the application of specific inequalities like the Arithmetic-Geometric Mean Inequality (AM-GM inequality). Let's break this down and check for any valid proof.

### Step-by-step Analysis:

1. **Squaring Both Sides**: Since \( x \) and \( y \) are non-negative, we can safely square both sides of the inequality without reversing it:

\[
\left( \frac{x}{2} \right)^2 \geq \left( \sqrt{xy + y} \right)^2
\]

This simplifies to:

\[
\frac{x^2}{4} \geq xy + y
\]

2. **Rearranging Terms**: Bring all terms to one side:

\[
\frac{x^2}{4} - xy - y \geq 0
\]

Multiply through by 4 to eliminate the denominator:

\[
x^2 - 4xy - 4y \geq 0
\]

3. **Factoring (if possible)**: We now analyze this quadratic inequality in terms of \( x \) and \( y \). This is where further algebraic manipulation or specific conditions on \( x \) and \( y \) might simplify the inequality.

Would you like me to continue this proof, or provide more details on any step?

### 5 Related Questions:
1. How can we apply the Arithmetic-Geometric Mean Inequality to this type of inequality?
2. Can we extend this inequality to complex numbers, or would it still hold only for real numbers?
3. What happens when \( y = 0 \)? How does the inequality simplify in that case?
4. Could this inequality be part of a larger mathematical concept such as convexity or optimization?
5. How would a graphical representation of this inequality look for various values of \( x \) and \( y \)?

**Tip**: When dealing with inequalities, always check if the equality case (equality sign) holds, as it can help identify critical points or conditions.

Let x and y be non-negative reals. Then, x/2 ≥ √(xy + y)

Explore how to solve the inequality x/2 ≥ √(xy + y) involving non-negative real numbers. The solution covers algebraic manipulation and the application of the Arithmetic-Geometric Mean Inequality. Suitable for students in Grades 10-12.

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