The value of x satisfying the inequalities (((2x−1)(x−1)^4 (x−2)^4 )/((x−2)(x−4)^4 ))≤ 0


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Question Number 72830 by Tip Top last updated on 03/Nov/19

$The value of x satisfying the$ $inequalities \frac{(2 x - 1) {(x - 1)}^{4} {(x - 2)}^{4}}{(x - 2) {(x - 4)}^{4}} ⩽ 0$
	Answered by MJS last updated on 03/Nov/19
	$defined for R\{2, 4} (((x−1)^4 (x−2)^4 )/((x−4)^4 ))≥0 ∀ x∈R\{4} ((2x−1)/(x−2))≤0 2x−1≥0∧x−2<0 ∨ 2x−1≤0∧x−2>0 ⇒ (1/2)≤x<2$
	$defined for R ∖ {2, 4}$ $\frac{{(x - 1)}^{4} {(x - 2)}^{4}}{{(x - 4)}^{4}} ⩾ 0 \forall x \in R ∖ {4}$ $\frac{2 x - 1}{x - 2} ⩽ 0$ $2 x - 1 ⩾ 0 \land x - 2 < 0 \lor 2 x - 1 ⩽ 0 \land x - 2 > 0$ $\Rightarrow \frac{1}{2} ⩽ x < 2$
	Answered by Tanmay chaudhury last updated on 03/Nov/19

	${(x - 1)}^{4}, {(x - 2)}^{4} a n d {(x - 4)}^{4} e v e n p o w e r$ $s o t h e y a r e a l w a y s + v e .$ $x \neq 2 x \neq 4$ $c r i t i c a l v a l u e o f x = \frac{1}{2} = 0.5 a n d 2$ $f (x) = \frac{2 x - 1}{(x - 2)} \times {[\frac{(x - 1) (x - 2)}{(x - 4)}]}^{4}$ $w h e n x > 2 f (x) > 0$ $w h e n x < 0.5 f (x) > 0$ $w h e n 2 > x > 0.5 f (x) < 0$ $s o l u t i o n x \in [0.5, 2) p l s c h e c k$
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