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

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Solution

Step 1 (Solve the inequation):
$4 x - 19 < \frac{3 x}{5} - 2 \leq \frac{- 2}{5} + x$
multiply both sides by 5
$20 x - 95 < 3 x - 10 \leq - 2 + 5 x 20 x - 95 < 3 x - 10 a n d 3 x - 10 \leq - 2 + 5 x 17 x < 85 a n d - 8 \leq 2 x x < 5 a n d x \geq - 4$
Step 2 (Representation on the number line):
The solution set will be $- 4 \leq x < 5$ , so the solution set on the number line will be
Hence the solution set of the inequality $4 x - 19 < \frac{3 x}{5} - 2 \leq \frac{- 2}{5} + x, \forall x \in R$ is ${x : - 4 \leq x < 5, \forall x \in R}$ and the representation of the solution set on the number line is as shown above.

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