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Question Number 167296 by mkam last updated on 12/Mar/22

$$\mathit{i}\mathit{f}\mid \mathit{x}-3\mid <2\mathit{t}\mathit{h}\mathit{e}\mathit{n}\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mid {\mathit{x}}^2-9\mid ⩽16$$

Commented bymkam last updated on 12/Mar/22

$$??????$$

Commented bycortano1 last updated on 12/Mar/22

$$\Rightarrow -2<\mathrm{x}-3<2$$ $$\Rightarrow -2+6<\mathrm{x}-3+6<2+6$$ $$\Rightarrow 4<\mathrm{x}+3<8$$ $$\{\begin{array}{l}-2<\mathrm{x}-3<2\\ 4<\mathrm{x}+3<8\end{array}\Rightarrow -16<(\mathrm{x}-3)(\mathrm{x}+3)<16$$ $$\Rightarrow -16<{\mathrm{x}}^2-9<16$$ $$\Rightarrow \mid {\mathrm{x}}^2-9\mid <16$$

Commented byMJS_new last updated on 12/Mar/22

$$\mathrm{I}\mathrm{beg}\mathrm{your}\mathrm{pardon},\mathrm{next}\mathrm{time}{\mathrm{I}}^{\prime }\mathrm{ll}\mathrm{set}\mathrm{my}\mathrm{alarm}$$ $$\mathrm{half}\mathrm{an}\mathrm{hour}\mathrm{earlier}\mathrm{so}\mathrm{that}\mathrm{I}\mathrm{can}\mathrm{answer}\mathrm{your}$$ $$\mathrm{questions}\mathrm{in}\mathrm{time}.$$

Answered by MJS_new last updated on 12/Mar/22

$$\mid x-3\mid <2$$ $$\Rightarrow 1<x<5$$ $$\Rightarrow 1<x^2<25$$ $$\Rightarrow -8<x^2-9<16$$ $$\Rightarrow \mid x^2-9\mid <16$$

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