Question

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

Answer 1

$2x-3<x+2\le 3x+5$

Let's break this up into two parts.....

First.......

$2x-3<x+2$ ........ (add 3 to both sides)

$2x<x+5$ ........ (subtract x from both sides)

$x<5$

Secondly.......

$x+2\le 3x+5$

(subtract 5 from both sides)

$x-3\le 3x$ ......... (subtract x from both sides)

$-3<=2x$ ....... (divide both sides by 2)

$-3\le 2x$ (which also means $x\ge -\frac{3}{2}$)

Thus.........$-\frac{3}{2}\le x<5$

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!!