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

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Answer 1 · 2016-06-21T09:23:22

Either $x<-3$ or $1 < x < 5$

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]}

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