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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{f}\left(\mathrm{x}\right)=\left|\mathrm{x}+\frac{1}{\mathrm{x}}\right|<4 \\ \mathrm{Case}\left(\mathrm{i}\right) \mathrm{When} \mathrm{x}>0, \mathrm{hence} \mathrm{mod} \mathrm{will} \mathrm{open} \mathrm{with} \mathrm{positive} \mathrm{sign}. \\ \Rightarrow \mathrm{x}+\frac{1}{\mathrm{x}}<4 \\ \Rightarrow \frac{{\mathrm{x}}^2+1}{\mathrm{x}}<4 \\ \Rightarrow {\mathrm{x}}^2+1<4\mathrm{x} \\ \Rightarrow {\mathrm{x}}^2-4\mathrm{x}+1<0 \\ \Rightarrow {\left({\mathrm{x}}^2-2\right)}^2-4+1<0 \\ \Rightarrow {\left({\mathrm{x}}^2-2\right)}^2-3<0 \\ \Rightarrow {\left({\mathrm{x}}^2-2\right)}^2-{\left(\sqrt{3}\right)}^2<0 \\ \Rightarrow \left[\mathrm{x}-2+\sqrt{3}\right]\left[\mathrm{x}-2-\sqrt{3}\right]<0 \\ \Rightarrow \left[\mathrm{x}-\left(2-\sqrt{3}\right)\right]\left[\mathrm{x}-\left(2+\sqrt{3}\right)\right]<0 \\ \Rightarrow \mathrm{x}\in \left[2-\sqrt{3},2+\sqrt{3}\right] \\ \mathrm{Case}\left(\mathrm{ii}\right) \mathrm{When} \mathrm{x}<0, \mathrm{hence} \mathrm{mod} \mathrm{will} \mathrm{open} \mathrm{with} \mathrm{negative} \mathrm{sign}. \\ -\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)<4 \\ \mathrm{Multiply} \mathrm{both} \mathrm{sides} \mathrm{by} -1 \\ \Rightarrow \frac{{\mathrm{x}}^2+1}{\mathrm{x}}>-4 \\ \mathrm{As} \mathrm{x}<0, \mathrm{hence} \mathrm{when} \mathrm{we} \mathrm{take} \mathrm{x} \mathrm{to} \mathrm{other} \mathrm{side}, \mathrm{sign} \mathrm{of} \mathrm{inequality} \mathrm{will} \mathrm{change}. \\ \Rightarrow {\mathrm{x}}^2+1<-4\mathrm{x} \\ \Rightarrow {\mathrm{x}}^2+4\mathrm{x}+1<0 \\ \Rightarrow {\left(\mathrm{x}+2\right)}^2-4+1<0 \\ \Rightarrow {\left(\mathrm{x}+2\right)}^2-{\left(\sqrt{3}\right)}^2<0 \\ \Rightarrow \left(\mathrm{x}+2+\sqrt{3}\right)\left(\mathrm{x}+2-\sqrt{3}\right)<0 \\ \Rightarrow \left[\mathrm{x}-\left(-2-\sqrt{3}\right)\right]\left[\mathrm{x}-\left(-2+\sqrt{3}\right)\right]<0 \\ \Rightarrow \mathrm{x}\in \left[-2-\sqrt{3},-2+\sqrt{3}\right] \\ \mathrm{Taking} \mathrm{union} \mathrm{of} \mathrm{two} \mathrm{cases} \mathrm{we} \mathrm{get} \\ \mathbf{x}\mathbf{\in }\left[\mathbf{2}\mathbf{-}\sqrt{\mathbf{3}}\mathbf{,}\mathbf{2}\mathbf{+}\sqrt{\mathbf{3}}\right]\mathbf{\cup }\left[\mathbf{-}\mathbf{2}\mathbf{-}\sqrt{\mathbf{3}}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{+}\sqrt{\mathbf{3}}\right] \end{gathered}$$

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