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When Kc is a very small number, we know that there will be a large amount of reactant and a small amount of product (Kc is products over reactant, so the larger the denominator the smaller Kc). When Kc is 10^-3 or smaller, we can assume that the change (x) is so small that is makes a negligible effect on the initial molarities of the reactants. This is when we can "ignore" x in an equilibrium calculation.
Today in lecture he mentioned that 10^-4 is the boundary, so anything 10^-4 and smaller can be approximated. However, he said that if K does equal 10^-4, you should double check your answer at the end (by plugging it back in) to see if there is a difference between your answer and the given because in some rare situations it could change the answer.
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