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Q.
The solution set of the inequality $| x + 2 | - | x -1|< x - \frac{3}{2}$ is
Linear Inequalities
Solution:
The inequality is $| x + 2 | - | x -1|< x - \frac{3}{2}$
Dividing the problem into three intervals :
(i) If $x < -2$, then
But $- \frac{3}{2} > - 2 $, hence no common values
$\Rightarrow \, x \in \phi$
(ii) If $-2 \leq x < 1$, then
$(x + 2) + (x -1) < x - \frac{3}{2} $
$ \Rightarrow \: x < - \frac{5}{2} $
But $- \frac{5}{2} < - 2 $, hence no common values
$ \Rightarrow \: x \, \in \, \phi$
(iii) If $x \geq 1$ , then
$ (x + 2) - (x -1) < x - \frac{3}{2} \, \Rightarrow x > \frac{9}{2} $
$\because \: \frac{9}{2} > 1 . \Rightarrow $ common solution is
$x > \frac{9}{2} \Rightarrow x \in \left( \frac{9}{2} , \infty \right)$
$\therefore $ Solution set is $x \in \left( \frac{9}{2} , \infty \right)$