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2-x-x-4-if-x-Z-2x-x-4-x-4-3-impossible-if-x-Z-x-x-1-2-x-x-3-x-1-1-x-lt-2-x-1-x-1-2-




Question Number 188430 by mnjuly1970 last updated on 01/Mar/23
      2⌊ x ⌋ − ⌊ −x ⌋ =4     −−−−    if  x∈Z ⇒  2x +x = 4 ⇒ x=(4/3)  ,impossible    if x∉ Z ⇒^(⌊−x⌋=−⌊x⌋−1) 2⌊x⌋+⌊x⌋=3       ⇒ ⌊ x ⌋= 1 ⇒  1≤ x < 2    ⇒^(x≠1)  x∈ (1 , 2)   ✓
$$ \\ $$$$\:\:\:\:\mathrm{2}\lfloor\:{x}\:\rfloor\:−\:\lfloor\:−{x}\:\rfloor\:=\mathrm{4} \\ $$$$\:\:\:−−−− \\ $$$$\:\:{if}\:\:{x}\in\mathbb{Z}\:\Rightarrow\:\:\mathrm{2}{x}\:+{x}\:=\:\mathrm{4}\:\Rightarrow\:{x}=\frac{\mathrm{4}}{\mathrm{3}}\:\:,{impossible} \\ $$$$\:\:{if}\:{x}\notin\:\mathbb{Z}\:\overset{\lfloor−{x}\rfloor=−\lfloor{x}\rfloor−\mathrm{1}} {\Rightarrow}\mathrm{2}\lfloor{x}\rfloor+\lfloor{x}\rfloor=\mathrm{3} \\ $$$$\:\:\:\:\:\Rightarrow\:\lfloor\:{x}\:\rfloor=\:\mathrm{1}\:\Rightarrow\:\:\mathrm{1}\leqslant\:{x}\:<\:\mathrm{2}\:\:\:\:\overset{{x}\neq\mathrm{1}} {\Rightarrow}\:{x}\in\:\left(\mathrm{1}\:,\:\mathrm{2}\right)\:\:\:\checkmark \\ $$$$ \\ $$

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