Multiplying or dividing both sides by a negative number changes the direction of an inequality. So we cannot multiply or divide both sides by a variable unless we know that the variable is either always positive or always negative. This is not a concern when solving equalities.
When applying a decreasing function to both sides of an inequality, the direction of the inequality changes.
Swapping both sides of an inequality also changes the direction of an inequality. Again we don't have to worry about this with equalities.
