Solve the following linear inequations and graph the solution set on a real number line.
$2 x - 7 < 5 x + 2 \leq 3 x + 14, x \in R$

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Solution

Finding the solution set of $x$
Given:
$2 x - 7 < 5 x + 2 \leq 3 x + 14, x \in R$
Consider $2 x - 7 < 5 x + 2$
By taking the like terms to one side of equation, we get,
$5 x - 2 x > - 7 - 2$
$3 x > - 9$
$x > - \frac{9}{3}$
$x > - 3$

Now, consider $5 x + 2 \leq 3 x + 14$
By taking the like terms to one side of equation,we get,

$5 x - 3 x \leq 14 - 2$
$2 x \leq 12$
$x \leq \frac{12}{2}$
$x \leq 6$
As per the condition in the question $x \in R$
Therefore, the solution set is $x = {- 3 < x \leq 6}$
Set can be represented in number line.

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