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the ice box should pretty much always be used because it can help get rid of mistakes and organize our thoughts. x is only negligible when the k value is ver, very small. it has to be less than 1x10^-3 in order to just ignore x, otherwise you can't simply get rid of it.
The X can only be taken out if it is very small and would not make a significant impact on the overall calculation, but this is usually only in either the numerator or denominator, rarely both. Therefore not all of the Xs can be considered negligible.
Professor Lavalle gave a great analogy about approximations. Basically, let's you have 1,000,000 dollars and you subtract it by x amount. If x is very small, like 10 dollars, then x is negligible because 1,000,000 - 10 = 999,990, which is almost the same as 1,000,000. In acids and bases, a weak acid like acetic acid will not dissociate much in water, so the change in acetic acid (x) is negligible.
The x is only negligible when adding/subtracting if the K value is very small (<10-3) or when the change in the concentrations/partial pressures is less than 5% of the original concentration/partial pressure. You will not always have a small K value, and the stoichiometric coefficients will not always be 1 which will sometimes make the changes in the concentrations/partial pressures be more than x (like 2x or 3x etc) and will require that value to be raised to a different power, so keeping everything in the ice box will allow you to keep track of everything and understand what goes where in the equation, and what value(s) you are solving for.
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