Solve the following inequation and graph the solution set on the number line. $2 x - 3 < x + 2 \leq 3 x + 5, x ϵ R$ .

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Solution

$2 x - 3 < x + 2 \leq 3 x + 5$
Let's break this up into two parts.....
First.......
$2 x - 3 < x + 2$ ........ (add 3 to both sides)
$2 x < x + 5$ ........ (subtract x from both sides)
$x < 5$
Secondly.......
$x + 2 \leq 3 x + 5$
(subtract 5 from both sides)
$x - 3 \leq 3 x$ ......... (subtract x from both sides)
$- 3 <= 2 x$ ....... (divide both sides by 2)
$- 3 \leq 2 x$ (which also means $x \geq - \frac{3}{2}$ )
Thus......... $- \frac{3}{2} \leq x < 5$
Or in interval notation...... $[- \frac{3}{2}, 5)$
Try picking some values "outside" this interval. You'll find that they won't satisfy both inequalities!!

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