Question

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

Answer 1

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]}