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Question Number 10463 by ABD last updated on 10/Feb/17

x∈Z  −1≤∣x−3∣<5 ⇒Σ=?

$${x}\in{Z} \\ $$ $$−\mathrm{1}\leqslant\mid{x}−\mathrm{3}\mid<\mathrm{5}\:\Rightarrow\Sigma=? \\ $$

Answered by mrW1 last updated on 10/Feb/17

−1≤∣x−3∣ is always true    ∣x−3∣<5  ⇔−5<x−3<5  ⇔−2<x<8  ⇔x=−1,0,1,2,3,4,5,6,7  Σx=(((−1+7)×9)/2)=27

$$−\mathrm{1}\leqslant\mid{x}−\mathrm{3}\mid\:{is}\:{always}\:{true} \\ $$ $$ \\ $$ $$\mid{x}−\mathrm{3}\mid<\mathrm{5} \\ $$ $$\Leftrightarrow−\mathrm{5}<{x}−\mathrm{3}<\mathrm{5} \\ $$ $$\Leftrightarrow−\mathrm{2}<{x}<\mathrm{8} \\ $$ $$\Leftrightarrow{x}=−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7} \\ $$ $$\Sigma{x}=\frac{\left(−\mathrm{1}+\mathrm{7}\right)×\mathrm{9}}{\mathrm{2}}=\mathrm{27} \\ $$

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