Question

Solve the following linear inequations and graph the solution set on a real number line
$-3\le \frac{1}{2}-\left(\frac{2x}{3}\right)\le 2\frac{2}{3},\text{x}\in N$

Answer 1

Finding the solution set of $x$
Given:
Consider $-3\le \frac{1}{2}-\left(\frac{2x}{3}\right)$
$-3\le \frac{(3-4x)}{6}$
$-18\le (3-4x)$
By taking the like terms to one side of equation, we get,
$-18-3\le -4x$
$-21\le -4x$
$x\le \frac{21}{4}$
$x=5\frac{1}{4}$
Now, consider $\frac{1}{2}-\left(\frac{2x}{3}\right)\le 2\frac{2}{3}$
By cross multiplication we get,
$3(3-4x)\le 48$
$9-12x\le 48$
By taking the like terms to one sided of equation,
we get,
$-12x\le 48-9$
$-12x\le 39$
$12x\ge -39$
$x\ge -\frac{39}{12}$
$x\ge -\frac{13}{4}=-3\frac{1}{4}$
As per the condition given in the equation, $x\in N$
Therefore, solution set $x=\{-3\frac{1}{4}\le x\le 5\frac{1}{4}\}=\{0,1,2,3,4,5\}$
Set can be represented in number line as