Why is N/(4-N)< 3 = N<3 incorrect?

Question

4/(N-4)<3

N<3(4-N)

N<12-3N

4N<12

N<3.

I know the final answer is x<3 or x>4, but why is how I just solved it wrong? And how do I reach x>4 and x<3 as a final answer?

Answer 1

You have to be really careful about inequalities.

4/(N-4)<3 can be written 4<3(N-4) and solved for N ONLY if N-4 is positive.

Assume N-4>0, that is, N>4 then 4<3N-12, 16<3N, 3N>16, N>16/3, N>5⅓. Since 5⅓>4, this solution for N is valid.

If N-4<0, the LHS becomes negative so, since all negative numbers must be less than any positive quantity (and 3 is positive), any value of N<4 will satisfy the inequality.

So the full solution is N<4 or N>5⅓.

Let's take an example: N=6, so N-4=2 and 4/(6-4)=2 which is less than 3. Note that N≠4.

If N=5, N-4=1 and 4/1=4 which is not less than 3, so the inequality fails if N lies strictly between 4 and 5⅓.