Solve the following inequation and represent the solution set on the number line:
$4 x - 19 < \frac{3 x}{5} - 2 \leq - \frac{2}{5} + x, x \in R$

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Solution

Take the first inequality:
$4 x - 19 < \frac{3 x}{5} - 2 \Rightarrow 20 x - 95 < 3 x - 10 \Rightarrow 17 x < 85$
$\Rightarrow x < 5 . . . (1)$

Now tale the second inequality:
$\frac{3 x}{5} - 2 \leq - \frac{2}{5} + x \Rightarrow 3 x - 10 \leq - 2 + 5 x \Rightarrow - 8 \leq 2 x$
$\Rightarrow x \geq - 4 . . . (2)$

From $(1)$ and $(2)$
$- 4 \leq x < 5$

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