How I thought about this is that I cancelled 1/x from both sides. On the left we then had 0 and on the right we had (x+1)/x. Now if I take a negative value of x here, quantity a becomes bigger.

is where you went wrong. If you “cancel” x from both sides by multiplying both quantities by x (which needs to be done very carefully due to inequalities flipping when x is negative and isn’t what I would recommend), you’ll get 1 as Quantity A, not 0.

If by “cancelling” you meant subtracting both quantities by 1/x, Quantity B would be