When dividing by a variable is considered, you must consider two cases separately:

First case: you have an EQUATION involved. In this case, you may divide whenever the variable cannot assume the zero value.

Example: \(ab - {b^3}c = d{b^2}\,\,\,\, \Rightarrow \,\,\,\,a - {b^2}c = db\) IF you can guarantee that b is different from zero.

Second case: you have an INEQUATION involved, that is, an inequality. In this case, you ALSO may divide whenever the variable cannot assume the zero value BUT

(i) if the variable can assume only positive values, the inequality sign is kept.

(ii) if the variable can assume only negative values, the inequality sign is reversed.

(If you cannot control the sign of the variable, you cannot be sure the inequality sign will be kept or reversed.)

Example: \(ab - {b^3}c > d{b^2}\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}

a - {b^2}c > db\,\,\,\,\,\,\left( {b > 0} \right) \hfill \\

a - {b^2}c < db\,\,\,\,\,\,\left( {b < 0} \right) \hfill \\

\end{gathered} \right.\)

Please note that it doesn´t matter if the variable takes only integer values or not. This is irrelevant for all the discussion above.

Regards,

fskilnik.

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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