Show how the assumption N>4 leads to a solution of the inequality that Neely did not find. (Hint: What is important about the value of 4-N?)

Question

N/(4-N) < 3

N<3(4 - N)

N < 12 - 3N

4N < 12

N < 3

6. Show how the assumption N>4 leads to a solution of the inequality that Neely did not find. (Hint: What is important about the value of 4-N?)

7. Now, write a complete solution of the inequality N/(4 - N) < 3

Answer 1

N/(4-N)<3. When we cross-multiply we can only do so if 4-N>0 otherwise the inequality is invalidated. So N must be less than 4 if we write N<3(4-N), N<12-3N, 4N<12, N<3. Since 3<4, then the assumption still holds. But this isn't the only solution. 3 is a positive number, so if the left-hand side were to be negative, the inequality would still be valid. Therefore 4-N<0, that is, N>4 also satisfies the inequality.

So the complete solution is N<3 or N>4. Note that if N<0, 4-N>0 but the numerator is negative so the fraction is negative, and the inequality holds. If N is between 3 and 4 (inclusive) the inequality fails; in fact, N=4 is an asymptote and the fraction is undefined.

You should now be able to answer the questions 6 and 7 from this info (the importance of 4-N and the complete solution).

See also:

https://www.mathhomeworkanswers.org/289908/why-is-n-4-n-3-n-3-incorrect