The number of integral solution of inequality $2 x - 3 < x + 2 \leq 3 x + 5$ is

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Solution

$2 x - 3 < x + 2 \leq 3 x + 5$
Split inequality into two parts
$2 x - 3 < x + 2$
and $x + 2 \leq 3 x + 5$

$2 x - 3 < x + 2$
$\Rightarrow x < 5 . . . (1)$

$x + 2 \leq 3 x + 5$
$\Rightarrow - 3 \leq 2 x$
$\Rightarrow x \geq - \frac{3}{2} . . . (2)$
From $(1)$ and $(2)$
$x \in [- \frac{3}{2}, 5)$
Integral values are ${- 1, 0, 1, 2, 3, 4}$

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