Solve for X: (5-x)·½ + 3x - 4(x-2) = ½(x+4) Linear Equation

To solve the equation $(5-x)\cdot\frac{1}{2} + 3x - 4(x-2) = \frac{1}{2}(x+4)$, we will follow a series of systematic steps.

Step 1: Simplify each side of the equation.

Distribute the terms:$<br> (5-x)\cdot\frac{1}{2} = \frac{5}{2} - \frac{x}{2}.$

Also distribute $-4(x-2)$:$<br> -4(x - 2) = -4x + 8 .$

Thus, the equation simplifies to:
$\frac{5}{2} - \frac{x}{2} + 3x - 4x + 8 = \frac{1}{2}(x + 4).$

Step 2: Combine like terms.

Combine terms on the left:
$\frac{5}{2} + 8 - \frac{x}{2} + 3x - 4x = \frac{1}{2}x + 2.$

Simplifying, we get:
$\frac{21}{2} - \frac{x}{2} - x = \frac{1}{2}x + 2.$

Step 3: Clear fractions and solve for $x$.

Multiply the entire equation by 2 to eliminate fractions:
$21 - x - 2x = x + 4.$

This simplifies to:
$21 - 3x = x + 4.$

Add $3x$ to both sides to isolate terms with $x$ on one side:
$21 = 4x + 4.$

Subtract 4 from both sides:
$17 = 4x.$

Finally, divide by 4:
$x = \frac{17}{4}.$

Thus, the solution is $x = \frac{17}{4}$.