If (x^2-4x+3)(x^2-3)<0(x2−4x+3)(x2−3)<0
then one (but not both) of the terms must be less than zero.
Case 1: (x^2-4x+3 < 0)(x2−4x+3<0) and (x^2-3 > 0)(x2−3>0)
color(white)("XXXX")XXXXx^2-4x+3 < 0x2−4x+3<0
color(white)("XXXX")XXXX(x-3)(x-1) < 0(x−3)(x−1)<0
color(white)("XXXX")XXXXagain one of the terms must be less than zero (and the other greater)
color(white)("XXXX")XXXXSince x-3 < x-1x−3<x−1
color(white)("XXXX")XXXXcolor(white)("XXXX")XXXXx-3 < 0x−3<0 and x-1 > 0x−1>0
color(white)("XXXX")XXXXcolor(white)("XXXX")XXXXx < 3x<3 and x > 1x>1
color(white)("XXXX")XXXXcolor(white)("XXXX")XXXXx epsilon (1,3)xε(1,3)
and
color(white)("XXXX")XXXXx^2-3>0x2−3>0
color(white)("XXXX")XXXXrarr x^2 > 3→x2>3
color(white)("XXXX")XXXXrArr x > sqrt(3)⇒x>√3
rArr sqrt(3) < x < 3⇒√3<x<3
Case 2: (x^2-4x+3 > 0)(x2−4x+3>0) and (x^2-3 < 0)(x2−3<0)
color(white)("XXXX")XXXXx^2-4x+3 > 0x2−4x+3>0
color(white)("XXXX")XXXX(x-3)(x-1) > 0(x−3)(x−1)>0
color(white)("XXXX")XXXXboth terms must be negative or both terms must be positive
color(white)("XXXX")XXXXand since x-3 < x-1x−3<x−1
color(white)("XXXX")XXXXcolor(white)("XXXX")XXXXx-3 > 0x−3>0 or x-1 < 0x−1<0
color(white)("XXXX")XXXXcolor(white)("XXXX")XXXXrArr x >3⇒x>3 or x < -1x<−1
and
color(white)("XXXX")XXXXx^2-3 < 0x2−3<0
color(white)("XXXX")XXXXrArr x^2 < 3⇒x2<3
color(white)("XXXX")XXXXrArr -sqrt(3) < x < sqrt(3)⇒−√3<x<√3
rArr -sqrt(3) < x < -1⇒−√3<x<−1
Combining: Case 1 or Case 2
-sqrt(3) < x < -1−√3<x<−1
or
sqrt(3) < x < 3√3<x<3