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If-0-lt-x-lt-1-then-x-1-2x-4-2x-1-




Question Number 182946 by myint last updated on 17/Dec/22
If  0 < x <1 ,  then  ∣ x −1 ∣ + ∣2x−4∣ + ∣2x+1∣=
$$\mathrm{If}\:\:\mathrm{0}\:<\:\mathrm{x}\:<\mathrm{1}\:,\:\:\mathrm{then}\:\:\mid\:\mathrm{x}\:−\mathrm{1}\:\mid\:+\:\mid\mathrm{2x}−\mathrm{4}\mid\:+\:\mid\mathrm{2x}+\mathrm{1}\mid= \\ $$
Commented by mr W last updated on 17/Dec/22
≠constant value!
$$\neq{constant}\:{value}! \\ $$
Answered by Frix last updated on 17/Dec/22
0<x<1 ⇒ x−1<0∧2x−4<0∧2x+1>0 ⇒  ∣x−1∣+∣2x−4∣+∣2x+1∣=  =−(x−1)−(2x−4)+2x+1=  =6−x
$$\mathrm{0}<{x}<\mathrm{1}\:\Rightarrow\:{x}−\mathrm{1}<\mathrm{0}\wedge\mathrm{2}{x}−\mathrm{4}<\mathrm{0}\wedge\mathrm{2}{x}+\mathrm{1}>\mathrm{0}\:\Rightarrow \\ $$$$\mid{x}−\mathrm{1}\mid+\mid\mathrm{2}{x}−\mathrm{4}\mid+\mid\mathrm{2}{x}+\mathrm{1}\mid= \\ $$$$=−\left({x}−\mathrm{1}\right)−\left(\mathrm{2}{x}−\mathrm{4}\right)+\mathrm{2}{x}+\mathrm{1}= \\ $$$$=\mathrm{6}−{x} \\ $$

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