find-the-solution-2x-x-2-x-3-

Question Number 84394 by jagoll last updated on 12/Mar/20

find the solution

\frac{2 x}{x - 2} ⩽ ∣ x - 3 ∣

Commented by john santu last updated on 12/Mar/20

(i) x ⩾ 3 \Rightarrow \frac{2 x}{x - 2} ⩽ x - 3

\Rightarrow 2 x ⩽ (x - 3) (x - 2)

\Rightarrow 2 x ⩽ x^{2} - 5 x + 6

\Rightarrow x^{2} - 7 x + 6 ⩾ 0, (x - 1) (x - 6) ⩾ 0

{x ⩽ 1 \lor x ⩾ 6} \land x ⩾ 3

∙) x ⩾ 6

(ii) x < 3 \Rightarrow \frac{2 x}{x - 2} ⩽ 3 - x

\Rightarrow \frac{2 x - (3 - x) (x - 2)}{x - 2} ⩽ 0

\frac{2 x - (- x^{2} + 5 x - 6)}{x - 2} ⩽ 0

\frac{x^{2} - 3 x + 6}{x - 2} ⩽ 0 \Rightarrow ∙) x < 2

the solution (- \infty, 2) \lor [6, \infty)

Commented by jagoll last updated on 12/Mar/20

thank you sir

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