Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 83674 by Power last updated on 05/Mar/20

Commented by Power last updated on 05/Mar/20

sir MJS

$$\mathrm{sir}\:\:\mathrm{MJS} \\ $$

Answered by mahdi last updated on 05/Mar/20

∣x∣+∣y∣≥∣x+y∣ ∣(x−1)+(x+1)∣+∣(x−2)+(x+2)∣+...+∣(x−500)+(x+500)∣ ≤∣x−1∣+∣x+1∣+...+∣x+500∣⇒ ∣2x∣+...+∣2x∣≤1000⇒ 500∣2x∣=1000∣x∣≤1000⇒∣x∣≤1 ⇒−1≤x≤1 1000=^? ∣x−1∣+∣x+1∣+...+∣x+500∣=^? (1−x)+(x+1)+...+(500−x)+(x+500) 1000≠2(((500×501)/2))⇒without answer

$$\mid\mathrm{x}\mid+\mid\mathrm{y}\mid\geqslant\mid\mathrm{x}+\mathrm{y}\mid \\ $$$$\mid\left(\mathrm{x}−\mathrm{1}\right)+\left(\mathrm{x}+\mathrm{1}\right)\mid+\mid\left(\mathrm{x}−\mathrm{2}\right)+\left(\mathrm{x}+\mathrm{2}\right)\mid+...+\mid\left(\mathrm{x}−\mathrm{500}\right)+\left(\mathrm{x}+\mathrm{500}\right)\mid \\ $$$$\leqslant\mid\mathrm{x}−\mathrm{1}\mid+\mid\mathrm{x}+\mathrm{1}\mid+...+\mid\mathrm{x}+\mathrm{500}\mid\Rightarrow \\ $$$$\mid\mathrm{2x}\mid+...+\mid\mathrm{2x}\mid\leqslant\mathrm{1000}\Rightarrow \\ $$$$\mathrm{500}\mid\mathrm{2x}\mid=\mathrm{1000}\mid\mathrm{x}\mid\leqslant\mathrm{1000}\Rightarrow\mid\mathrm{x}\mid\leqslant\mathrm{1} \\ $$$$\Rightarrow−\mathrm{1}\leqslant\mathrm{x}\leqslant\mathrm{1} \\ $$$$\mathrm{1000}\overset{?} {=}\mid\mathrm{x}−\mathrm{1}\mid+\mid\mathrm{x}+\mathrm{1}\mid+...+\mid\mathrm{x}+\mathrm{500}\mid\overset{?} {=}\left(\mathrm{1}−\mathrm{x}\right)+\left(\mathrm{x}+\mathrm{1}\right)+...+\left(\mathrm{500}−\mathrm{x}\right)+\left(\mathrm{x}+\mathrm{500}\right) \\ $$$$\mathrm{1000}\neq\mathrm{2}\left(\frac{\mathrm{500}×\mathrm{501}}{\mathrm{2}}\right)\Rightarrow\mathrm{without}\:\mathrm{answer} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com