The book says that when you multiply/divide both sides of an inequality, the relation switches if what you're multiplying/dividing by is negative. That's fine if I'm multiplying/dividing by a constant, but what happens when I'm multiplying/dividing by a variable or an expression containing variables? Does the inequality then branch into two forms?

For example, let's say I have the inequality 5 < 10 and I want to multiply both sides by x-1. Does the inequality then branch out to multiple inequalities?

* 5\*(x-1) < 10\*(x-1), if x-1 > 0

* 5\*(x-1) > 10\*(x-1), if x-1 < 0

* ????, if x-1 = 0

Then what happens when you keep taking on stuff to the sides? Now lets say I want to divide both sides by y. Do the branches keep exploding out?

* (5\*(x-1))/y < (10\*(x-1))/y, if x-1 > 0 and y > 0

* (5\*(x-1))/y > (10\*(x-1))/y, if x-1 > 0 and y < 0

* (5\*(x-1))/y > (10\*(x-1))/y, if x-1 < 0 and y > 0

* (5\*(x-1))/y < (10\*(x-1))/y, if x-1 < 0 and y < 0

* ... cases where x-1 and y are 0 ...