The number of positive integral solutions of the in-equation #(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0# is ?

Ans : 2

1 Answer
Narad T.
Feb 25, 2018

The solution is #x in x in [4/3,2]#

Explanation:

Let #f(x)=(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)#

Therere are #2# vertical asymptotes

Let 's build the sign chart

#color(white)(aaa)##x##color(white)(aaa)##-oo##color(white)(aaaa)##0##color(white)(aaaaa)##4/3##color(white)(aaaa)##2##color(white)(aaaa)##7/2##color(white)(aaaaa)##5##color(white)(aaaa)##+oo#

#color(white)(aaa)##x^2##color(white)(aaaaaa)##+##color(white)(aa)##0##color(white)(a)##+##color(white)(aaa)##+##color(white)(aa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)()##(3x-4)^3##color(white)(aaaa)##-##color(white)(aaa)##-##color(white)(a)##0##color(white)(a)##+##color(white)(aa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)()##(x-2)^4##color(white)(aaaaa)##+##color(white)(aaa)##+##color(white)(aaa)##+##color(white)(a)##0##color(white)(a)##+##color(white)(aaa)##+##color(white)(aaaa)##+#

#color(white)()##(2x-7)^6##color(white)(aaaa)##+##color(white)(aaa)##+##color(white)(aaa)##+##color(white)(a)####color(white)(aa)##+##color(white)(a)##||##color(white)(a)##+##color(white)(aaaa)##+#

#color(white)()##(x-5)^5##color(white)(aaaaa)##-##color(white)(aaa)##-##color(white)(aa)##-##color(white)(a)####color(white)(aaa)##+##color(white)(a)####color(white)(a)##+##color(white)(aa)##||##color(white)(aa)##+#

#color(white)()##f(x)##color(white)(aaaaaaaa)##+##color(white)(aaa)##+##color(white)(aa)##-##color(white)(a)####color(white)(aaa)##+##color(white)(a)##||##color(white)(a)##+##color(white)(aa)##||##color(white)(aa)##+#

Therefore,

#f(x)<=0# when #x in [4/3,2]#

graph{(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6) [-36.53, 36.56, -18.27, 18.25]}

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