How do you solve and write the following in interval notation: (x + 3)(x – 1)(x – 5) < 0?

1 Answer
Jun 21, 2016

Either x<-3 or 1 < x < 5

Explanation:

If (x+3)(x-1)(x-5)<0 (i.e. the product is negative) then options are

(i) All three are negative i.e. x+3<0 and x-1<0 and x-5<0 i.e. x<-3 and x<1 and x<5. Hence x<-3.

(ii) While x+3>0 and x-1>0, x-5<0 i.e. x>-3 and x>1 and x<5. This is possible only if 1<x<5.

(iii) While x+3>0 and x-5>0, x-1<0 i.e. x>-3 and x>5 and x<1. This is just not possible.

(iv) While x-1>0 and x-5>0, x+3<0 i.e. x>1 and x>5 and x<-3. This is too not possible.

(v) It is also not possible to have all positive.

Hence the solution is either x<-3 or 1 < x < 5.

This is also apparent from the following graph.

graph{(x+3)(x-1)(x-5) [-10, 10, -40, 40]}