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]}