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No what the reaction quotient (Q) is supposed to tell you is where the how the reaction will shift to reach equilibrium. If Q>K then equilibrium will shift towards the left. So when you put it into the ice chart you will put +x on the change for the reactants and will put -x for the change in the products. If Q<K then the reaction will shift toward the right and you will put -x for the change in the reactants and +x for the change in the products.
It's not that the equation flips from products becoming the reactants, but more that we are trying to calculate at which concentrations the reaction reaches equilibrium. If you just flip the equilibrium constant to 1/K, you're not changing anything about the concentrations.
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