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Question Number 100130 by bemath last updated on 25/Jun/20
how many integer number   satisfy the equation   ((sin x−∣x+2∣)/(x^2 −4x−5)) ≥ 0
$$\mathrm{how}\:\mathrm{many}\:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\frac{\mathrm{sin}\:\mathrm{x}−\mid\mathrm{x}+\mathrm{2}\mid}{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}−\mathrm{5}}\:\geqslant\:\mathrm{0}\: \\ $$
Answered by mr W last updated on 25/Jun/20
let f(x)=sin x −∣x+2∣  for x≤−2:  f(x)=sin x+x+2  f′(x)=cos x+1≥0 ⇒increasing  f(x)≤f(−2)=sin (−2)≈−0.909    for x≥−2:  f(x)=sin x−x−2  f′(x)=cos x−1≤0 ⇒decreasing  f(x)≤f(−2)=sin (−2)≈−0.909    ⇒f(x)=sin x−∣x+2∣≤sin (−2)≈−0.909<0    ((sin x−∣x+2∣)/(x^2 −4x−5)) ≥ 0   since always sin x−∣x+2∣<0  ⇒x^2 −4x−5<0  i.e. (x+1)(x−5)<0  ⇒−1<x<5  ⇒x=0,1,2,3,4 ⇒5 integers
$${let}\:{f}\left({x}\right)=\mathrm{sin}\:{x}\:−\mid{x}+\mathrm{2}\mid \\ $$$${for}\:{x}\leqslant−\mathrm{2}: \\ $$$${f}\left({x}\right)=\mathrm{sin}\:{x}+{x}+\mathrm{2} \\ $$$${f}'\left({x}\right)=\mathrm{cos}\:{x}+\mathrm{1}\geqslant\mathrm{0}\:\Rightarrow{increasing} \\ $$$${f}\left({x}\right)\leqslant{f}\left(−\mathrm{2}\right)=\mathrm{sin}\:\left(−\mathrm{2}\right)\approx−\mathrm{0}.\mathrm{909} \\ $$$$ \\ $$$${for}\:{x}\geqslant−\mathrm{2}: \\ $$$${f}\left({x}\right)=\mathrm{sin}\:{x}−{x}−\mathrm{2} \\ $$$${f}'\left({x}\right)=\mathrm{cos}\:{x}−\mathrm{1}\leqslant\mathrm{0}\:\Rightarrow{decreasing} \\ $$$${f}\left({x}\right)\leqslant{f}\left(−\mathrm{2}\right)=\mathrm{sin}\:\left(−\mathrm{2}\right)\approx−\mathrm{0}.\mathrm{909} \\ $$$$ \\ $$$$\Rightarrow{f}\left({x}\right)=\mathrm{sin}\:{x}−\mid{x}+\mathrm{2}\mid\leqslant\mathrm{sin}\:\left(−\mathrm{2}\right)\approx−\mathrm{0}.\mathrm{909}<\mathrm{0} \\ $$$$ \\ $$$$\frac{\mathrm{sin}\:\mathrm{x}−\mid\mathrm{x}+\mathrm{2}\mid}{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}−\mathrm{5}}\:\geqslant\:\mathrm{0}\: \\ $$$${since}\:{always}\:\mathrm{sin}\:{x}−\mid{x}+\mathrm{2}\mid<\mathrm{0} \\ $$$$\Rightarrow{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{5}<\mathrm{0} \\ $$$${i}.{e}.\:\left({x}+\mathrm{1}\right)\left({x}−\mathrm{5}\right)<\mathrm{0} \\ $$$$\Rightarrow−\mathrm{1}<{x}<\mathrm{5} \\ $$$$\Rightarrow{x}=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\Rightarrow\mathrm{5}\:{integers} \\ $$
Commented by Rasheed.Sindhi last updated on 25/Jun/20
T_(H_A  N) k_(S  )       S_I  R        !
$$\mathcal{T}_{\mathcal{H}_{\mathcal{A}} \:\mathcal{N}} {k}_{\mathcal{S}\:\:} \: \\ $$$$\:\:\:\mathcal{S}_{\mathcal{I}} \:\mathcal{R} \\ $$$$\:\:\:\:\:\:! \\ $$
Commented by bobhans last updated on 26/Jun/20
cooll great
$$\mathrm{cooll}\:\mathrm{great} \\ $$

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