To prove the inequality:
$x < \frac{2a + 2b}{2}$ we simplify the right-hand side:

$\frac{2a + 2b}{2} = \frac{2(a + b)}{2} = a + b.$

Thus, the inequality reduces to:

$x < a + b.$

To prove this, we need more context. Could you clarify the conditions on $x$, $a$, and $b$? For example:

• Is there a given relationship between $x$, $a$, and $b$?
• Are $a$ and $b$ positive numbers?
• Is $x$ defined by another equation or inequality?

Once we have this information, we can construct a formal proof.

Would you like me to assume $x < a + b$ as given and justify it algebraically? Or do you have additional constraints to consider?

Related Questions:

1. How does the inequality change if $a$ and $b$ are negative?
2. What happens if we assume $x = a + b$ instead of $x < a + b$?
3. Can this inequality be extended to other forms, such as for three variables?
4. How does this inequality relate to the arithmetic mean?
5. What real-world applications does this inequality have?

Tip:

When proving inequalities, always check for missing assumptions. Small details can change the entire proof!