Algebra Guides: Extra Inequalities - pg. 169, #4

[JohnnyL37]

Problem: (4/x) < (-1/3); what is the possible range of values for x?

I understand the question beforehand (#3) which is similar. However, for #4, when setting up the inequality for Case 2 (x is negative), I'm a little bit confused on how to solve for x.

The book shows:

(4/x) < (-1/3)
12 > -x
-12 < x

I'm curious: when you cross multiply, why have you already flipped the inequality sign before you divided by the negative x?

For instance, why isn't it this instead:

(4/x) < (-1/3)
12 < -x
-12 > x

I know you can check your answer against the answer given in the book to confirm that x must indeed be greater than -12. I guess my problem is that I don't know how to properly cross multiply with an inequality.

Any help?

[RonPurewal]

"cross-multiply" is not actually a thing.

"cross-multiply" is, in reality, "multiply by the product of both denominators".

try it; you'll see.
e.g., if you take a/b = c/d and multiply both sides by bd, you'll get the familiar "cross-products" ad = bc.

so, the first step of this problem is not "cross-multiply", because "cross-multiply" does not exist.
instead, the first step of this work-up is "multiply both sides by 3x". since x has to be negative, the inequality must be flipped.

[RonPurewal]

also, make sure you understand why x has to be negative.

specifically, we know that 4/x is less than a negative number (-1/3). so, 4/x is also negative. and, if 4/x is negative, then so is x.