Stuart Kovinsky wrote: Remember, we can do whatever we want to an inequality as long as we do the same thing to both sides. As long as we don't multiply or divide by a negative, inequalities follow the exact same rules as equations.

No, that is not true- inequalities certainly do not 'follow the exact same rules as equations' with the one exception of multiplying or dividing by negatives. You cannot take reciprocals on both sides of an inequality. You cannot square both sides of an inequality. You cannot multiply two inequalities together. You cannot take the absolute value of both sides of an inequality. You cannot multiply both sides of an inequality by zero. I could go on. In each case, you risk arriving at something that is incorrect, unless you use additional information about the quantities on either side of the inequality. You can safely do all of these things with equations (with the lone exception of taking reciprocals when each side is equal to zero).

With an inequality, it is always safe to do the following: you can add or subtract the same quantitiy from both sides. You can multiply or divide by the same positive quantitiy on both sides. You can multiply or divide by the same negative quantity on both sides if you reverse the inequality. In other situations, we need to be more careful.