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Yes you do, you also flip the inequality sign if you multiply by a negative # The > and < signs are strictly the "Greater than" and "Less than" signs. The inequality sign is an = with a / stroke through it. If you divide an inequality by -1 it remains an inequality.
No, inequalities have more than, less then, at least, or no more than signs. Equations just have equal signs. An inequality answer can't be written as just a number it has to have a sign with it.
No. An inequality will show two expressions that aren't equal.
It is still an inequality but not a new inequality. It will not alter the existence or non-existence of a solution to a system of linear equations / inequalities.
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