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  4. If |2x-3|<|x+5|, then x lies in the interval

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Q. If $ |2x-3|<|x+5|, $ then $ x $ lies in the interval

KEAMKEAM 2011Linear Inequalities

Solution:

$ |2x-3|<|x+5| $
$ \Rightarrow $ $ \pm (2x-3)<\pm (x+5) $ ?.. (i) First we take positive sign and negative sign on both sides,
$ \Rightarrow $ $ 2x-3 $ \Rightarrow $ $ x<8 $ ?.(ii) $ -(2x-3)<-(x+5) $
$ \Rightarrow $ $ 2x-3>x+5 $
$ \Rightarrow $ $ x>8 $ ..(iii)
Now, we take negative sign in RHS and positive in LHS,
$ \Rightarrow $ $ 2x-3<-x-5 $
$ \Rightarrow $ $ 3x<-2 $
$ \Rightarrow $ $ x<-\frac{2}{3} $ .. (iv)
and we take negative sign LHS and positive in RHS,
$ \Rightarrow $ $ -(2x-3) $ \Rightarrow $ $ -2x+3 $ \Rightarrow $ $ -2 < 3x $
$ \Rightarrow $ $ -2/3 < x $ ?.(v)
From Eqs. (ii) and (v), $ -2/3 < x < 8 $
$ \Rightarrow $ $ x\in (-2/3,8) $
From Eqs. (iii) and (iv), $ x\in (-\infty ,-2/3)\cup (8,\infty ) $