Simplifying the Algebraic Expression

We are asked to simplify the given algebraic expression:

$$ E = \frac{{\left( {x + y} \right) - \left| {x - y} \right|}}{2} $$

To simplify this expression, we need to consider the definition of the absolute value function, \( |a| \), which is:

• \( |a| = a \) if \( a \ge 0 \)
• \( |a| = -a \) if \( a < 0 \)

In our expression, the absolute value term is \( |x - y| \). We will analyze this based on the relationship between \( x \) and \( y \).

Analyzing Cases for Absolute Value

Case 1: \( x \ge y \)

If \( x \) is greater than or equal to \( y \), then \( x - y \ge 0 \). According to the definition of absolute value:

$$ |x - y| = x - y $$

Now, substitute this back into the original expression \( E \):

$$ E = \frac{{\left( {x + y} \right) - \left( {x - y} \right)}}{2} $$

Simplify the numerator:

$$ E = \frac{x + y - x + y}{2} $$

$$ E = \frac{2y}{2} $$

$$ E = y $$

Since we assumed \( x \ge y \), the value \( y \) represents the minimum of \( x \) and \( y \). Thus, in this case, \( E = \min(x, y) \).

Case 2: \( x < y \)

If \( x \) is less than \( y \), then \( x - y < 0 \). According to the definition of absolute value:

$$ |x - y| = -(x - y) = y - x $$

Now, substitute this back into the original expression \( E \):

$$ E = \frac{{\left( {x + y} \right) - \left( {y - x} \right)}}{2} $$

Simplify the numerator:

$$ E = \frac{x + y - y + x}{2} $$

$$ E = \frac{2x}{2} $$

$$ E = x $$

Since we assumed \( x < y \), the value \( x \) represents the minimum of \( x \) and \( y \). Thus, in this case, \( E = \min(x, y) \).

Conclusion: Minimum Value Identification

In both possible cases (\( x \ge y \) and \( x < y \)), the expression simplifies to the minimum value between \( x \) and \( y \).

Therefore, the expression \( \frac{{\left( {x + y} \right) - \left| {x - y} \right|}}{2} \) is equal to the minimum of \( x \) and \( y \).

This corresponds to Option 2.