|x+2|=2*(3-x)

To solve the equation \(|x + 2| = 2(3 - x)\), we can break it into two cases based on the definition of absolute value.

Case 1: \(x + 2 \geq 0\) (i.e., \(x \geq -2\))

In this case, \(|x + 2| = x + 2\). The equation becomes: \[ x + 2 = 2(3 - x) \] Now, simplify the right side: \[ x + 2 = 6 - 2x \] Next, add \(2x\) to both sides: \[ x + 2 + 2x = 6 \] \[ 3x + 2 = 6 \] Now, subtract 2 from both sides: \[ 3x = 4 \] Divide by 3: \[ x = \frac{4}{3} \]

Since \(\frac{4}{3} \geq -2\), this solution is valid.

Case 2: \(x + 2 < 0\) (i.e., \(x < -2\))

In this case, \(|x + 2| = -(x + 2)\). The equation becomes: \[ -(x + 2) = 2(3 - x) \] This simplifies to: \[ -x - 2 = 6 - 2x \] Adding \(2x\) to both sides gives: \[ -x + 2x - 2 = 6 \] \[ x - 2 = 6 \] Adding 2 to both sides: \[ x = 8 \]

However, \(8\) is not less than \(-2\), so this solution is invalid.

Final Solution

The only valid solution is: \[ x = \frac{4}{3} \]