To solve the equation \( 2(6 - 2x) + 3x = 4 + 2(5 + x) \), we will start by expanding both sides.
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Distribute on the left side: \[ 2(6) - 2(2x) + 3x = 12 - 4x + 3x = 12 - x \]
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Distribute on the right side: \[ 4 + 2(5) + 2(x) = 4 + 10 + 2x = 14 + 2x \]
Now, our equation looks like this: \[ 12 - x = 14 + 2x \]
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Next, we isolate \( x \) by adding \( x \) to both sides: \[ 12 = 14 + 3x \]
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Then, subtract \( 14 \) from both sides: \[ 12 - 14 = 3x \implies -2 = 3x \]
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Finally, divide both sides by \( 3 \): \[ x = -\frac{2}{3} \]
Thus, the solution is: \[ \boxed{-\frac{2}{3}} \]