The number of positive integral solutions of the in-equation $\frac{x^{2} {(3 x - 4)}^{3} {(x - 2)}^{4}}{{(x - 5)}^{5} {(2 x - 7)}^{6}} \leq 0$ $(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0$ is ?

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The number of positive integral solutions of the in-equation $\frac{x^{2} {(3 x - 4)}^{3} {(x - 2)}^{4}}{{(x - 5)}^{5} {(2 x - 7)}^{6}} \leq 0$ $(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0$ is ?

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Answer 1 · 2018-02-25T13:46:33

The solution is $x \in x \in [\frac{4}{3}, 2]$ $x in x in [4/3,2]$

Explanation:

Let $f (x) = \frac{x^{2} {(3 x - 4)}^{3} {(x - 2)}^{4}}{{(x - 5)}^{5} {(2 x - 7)}^{6}}$ $f(x)=(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)$

Therere are $2$ $2$ vertical asymptotes

Let 's build the sign chart

$a a a$ $color(white)(aaa)$ $x$ $x$ $a a a$ $color(white)(aaa)$ $- \infty$ $-oo$ $a a a a$ $color(white)(aaaa)$ $0$ $0$ $a a a a a$ $color(white)(aaaaa)$ $\frac{4}{3}$ $4/3$ $a a a a$ $color(white)(aaaa)$ $2$ $2$ $a a a a$ $color(white)(aaaa)$ $\frac{7}{2}$ $7/2$ $a a a a a$ $color(white)(aaaaa)$ $5$ $5$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a$ $color(white)(aaa)$ $x^{2}$ $x^2$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$ $a a$ $color(white)(aa)$ $0$ $0$ $a$ $color(white)(a)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a$ $color(white)(aa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

$color(white)()$ ${(3 x - 4)}^{3}$ $(3x-4)^3$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a$ $color(white)(aaa)$ $-$ $-$ $a$ $color(white)(a)$ $0$ $0$ $a$ $color(white)(a)$ $+$ $+$ $a a$ $color(white)(aa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

$color(white)()$ ${(x - 2)}^{4}$ $(x-2)^4$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a$ $color(white)(a)$ $0$ $0$ $a$ $color(white)(a)$ $+$ $+$ $a a a$ $color(white)(aaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

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

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

$color(white)()$ $f(x)$ $color(white)(aaaaaaaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(aa)$ $-$ $color(white)(a)$ #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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The number of positive integral solutions of the in-equation $\frac{x^{2} {(3 x - 4)}^{3} {(x - 2)}^{4}}{{(x - 5)}^{5} {(2 x - 7)}^{6}} \leq 0$ $(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0$ is ?

Ans : 2

1 Answer

Explanation:

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The number of positive integral solutions of the in-equation x2(3x−4)3(x−2)4(x−5)5(2x−7)6≤0(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0 is ?

Ans : 2

1 Answer

Explanation:

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The number of positive integral solutions of the in-equation $\frac{x^{2} {(3 x - 4)}^{3} {(x - 2)}^{4}}{{(x - 5)}^{5} {(2 x - 7)}^{6}} \leq 0$ $(x^2(3x-4)^3(x-2)^4)/((x-5)^5(2x-7)^6)<=0$ is ?