find the greatest integral x which satisfied the system of inequations.
$0.5 (2 x - 5) > \frac{2 - x}{2} + 1$
$0.2 (3 x - 2) + 3 > \frac{4 x}{3} - 0.5 (x - 1)$

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Solution

Given system of inequalities
$0.5 (2 x - 5) > \frac{2 - x}{2} + 1$ ....(1)
$0.2 (3 x - 2) + 3 > \frac{4 x}{3} - 0.5 (x - 1)$
Inequality (1) can be written as,
$\frac{1}{2} (2 x - 5) > \frac{4 - x}{2}$
$\Rightarrow 4 x - 10 > 8 - 2 x$
$\Rightarrow x > 3$ ....(3)
Inequality (2) can be written as
$\frac{1}{5} (3 x - 2) + 3 > \frac{4 x}{3} - \frac{1}{2} (x - 1)$
$\Rightarrow 18 x + 78 > 25 x + 15$
$\Rightarrow x < 9$ ....(4)
From (3) and (4),
$3 < x < 9$
So, the greatest integral solution is 8.

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