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Question Number 16086 by Tinkutara last updated on 17/Jun/17
![The number of values of x which are satisfying the equation ∣x + 4∣ = 8[x] + x − 4 is? (where [∙] Greatest Integer Function)](Q16086.png)
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation}\:\mid{x}\:+\:\mathrm{4}\mid\:=\:\mathrm{8}\left[{x}\right] \\ $$$$+\:{x}\:−\:\mathrm{4}\:\mathrm{is}?\:\left(\mathrm{where}\:\left[\centerdot\right]\:\mathrm{Greatest}\:\mathrm{Integer}\right. \\ $$$$\left.\mathrm{Function}\right) \\ $$
Commented by prakash jain last updated on 17/Jun/17
![∣x+4∣=8[x]+x−4 x≥−4 [x]+{x}+4=8[x]+[x]+{x}−4 8[x]=8⇒[x]=1 solution 1≤x<2 check x=1.5 5.5=8+1.5−4 x<−4 −(x+4)=8[x]+x−4 −x−4=8[x]+x−4 −2x=8[x] x=−4[x] [x]+{x}=−4[x] {x}=−5[x] no solution for x<−4 solution set1≤x<2](Q16098.png)
$$\mid{x}+\mathrm{4}\mid=\mathrm{8}\left[{x}\right]+{x}−\mathrm{4} \\ $$$${x}\geqslant−\mathrm{4} \\ $$$$\left[{x}\right]+\left\{{x}\right\}+\mathrm{4}=\mathrm{8}\left[{x}\right]+\left[{x}\right]+\left\{{x}\right\}−\mathrm{4} \\ $$$$\mathrm{8}\left[{x}\right]=\mathrm{8}\Rightarrow\left[{x}\right]=\mathrm{1} \\ $$$${solution}\:\mathrm{1}\leqslant{x}<\mathrm{2} \\ $$$${check} \\ $$$${x}=\mathrm{1}.\mathrm{5} \\ $$$$\mathrm{5}.\mathrm{5}=\mathrm{8}+\mathrm{1}.\mathrm{5}−\mathrm{4} \\ $$$${x}<−\mathrm{4} \\ $$$$−\left({x}+\mathrm{4}\right)=\mathrm{8}\left[{x}\right]+{x}−\mathrm{4} \\ $$$$−{x}−\mathrm{4}=\mathrm{8}\left[{x}\right]+{x}−\mathrm{4} \\ $$$$−\mathrm{2}{x}=\mathrm{8}\left[{x}\right] \\ $$$${x}=−\mathrm{4}\left[{x}\right] \\ $$$$\left[{x}\right]+\left\{{x}\right\}=−\mathrm{4}\left[{x}\right] \\ $$$$\left\{{x}\right\}=−\mathrm{5}\left[{x}\right] \\ $$$${no}\:{solution}\:{for}\:{x}<−\mathrm{4} \\ $$$${solution}\:{set}\mathrm{1}\leqslant{x}<\mathrm{2} \\ $$