1. ## need help inequality

please show me the plotting sets involved how you get them and explain the graph as draw visullly thanks for your help
x+ 2y<=3
2x-3y<=6
solve the following system of lnear inequalities by graphing

2. Hello, lizard4!

Okay, here it is in baby-steps . . .

Solve by graphing: . $\begin{array}{ccc}x+ 2y & \leq & 3 \\2x-3y & \leq & 6 \end{array}$

Graph the first inequality . . .

Graph the line: . $x + 2y \:=\:3$
. . It has intercepts: $(3,\,0),\;\left(0,\,\frac{3}{2}\right)$. .Draw the line.
Solve the inequality for $y\!:\;\;2y \:\leq \:-x + 3\quad\Rightarrow\quad y \:\leq\:-\frac{1}{2}x + \frac{3}{2}$
Since it is " $\leq$", shade the region below the line.
Code:
            |
*     |
:::*  |
::::::o 3/2
::::::|::*
::::::|:::::*   3
---:-:-:+:-:-:-:-:o----------------
::::::|::::::::::::*
::::::|:::::::::::::::*
::::::|::::::::::::::::::*
::::::|:::::::::::::::::::::::*
||

Graph the second inequality . . .

Graph the line: . $2x - 3y \:=\:6$
. . It has intercepts: . $(3\,0),\;(0,-2)$. .Draw the line.
Solve the inequality for $y\!:\;\;-3y \:\leq\:-2x + 6\quad\Rightarrow\quad y \:\geq \:\frac{2}{3}x - 2$
. . (When multiplying or dividing by a negative,
. . . reverse the inequality . . . remember?]

Since it is " $\geq$", shade the region above the line.
Code:
            |
::::::|:::::::::::::::::::*
::::::|:::::::::::::::::*
::::::|:::::::::::::::*
::::::|:::::::::::::*
::::::|:::::::::::*
---:-:-:+:-:-:-:-:o----------------
::::::|:::::::* 3
::::::|:::::*
::::::|:::*
::::::|:*
::::::o -2
::::* |
::*   |
*     |

The solution is the region that has been shaded twice.
Code:
            |
*     |                   *
:::*  |                 *
::::::o 3/2           *
::::::|::*          *
::::::|:::::*     *
---:-:-:+:-:-:-:-:o----------------
::::::|:::::::* 3  *
::::::|:::::*         *
::::::|:::*              *
::::::|:*                    *
::::::o -2
::::* |
::*   |
*     |

3. Originally Posted by Soroban
Hello, lizard4!

Okay, here it is in baby-steps . . .

Graph the first inequality . . .

Graph the line: . $x + 2y \:=\:3$
. . It has intercepts: $(3,\,0),\;\left(0,\,\frac{3}{2}\right)$. .Draw the line.
Solve the inequality for $y\!:\;\;2y \:\leq \:-x + 3\quad\Rightarrow\quad y \:\leq\:-\frac{1}{2}x + \frac{3}{2}$
Since it is " $\leq$", shade the region below the line.
Code:
            |
*     |
:::*  |
::::::o 3/2
::::::|::*
::::::|:::::*   3
---:-:-:+:-:-:-:-:o----------------
::::::|::::::::::::*
::::::|:::::::::::::::*
::::::|::::::::::::::::::*
::::::|:::::::::::::::::::::::*
||

Graph the second inequality . . .

Graph the line: . $2x - 3y \:=\:6$
. . It has intercepts: . $(3\,0),\;(0,-2)$. .Draw the line.
Solve the inequality for $y\!:\;\;-3y \:\leq\:-2x + 6\quad\Rightarrow\quad y \:\geq \:\frac{2}{3}x - 2$
. . (When multiplying or dividing by a negative,
. . . reverse the inequality . . . remember?]

Since it is " $\geq$", shade the region above the line.
Code:
            |
::::::|:::::::::::::::::::*
::::::|:::::::::::::::::*
::::::|:::::::::::::::*
::::::|:::::::::::::*
::::::|:::::::::::*
---:-:-:+:-:-:-:-:o----------------
::::::|:::::::* 3
::::::|:::::*
::::::|:::*
::::::|:*
::::::o -2
::::* |
::*   |
*     |

The solution is the region that has been shaded twice.
Code:
            |
*     |                   *
:::*  |                 *
::::::o 3/2           *
::::::|::*          *
::::::|:::::*     *
---:-:-:+:-:-:-:-:o----------------
::::::|:::::::* 3  *
::::::|:::::*         *
::::::|:::*              *
::::::|:*                    *
::::::o -2
::::* |
::*   |
*     |
How long did it take you to draw these?! You're really patient today to be drawing all these intricate diagrams like that

4. Well, there hasn't been too much traffic at my favorite math-help sites today.
. . And this is what I do when I'm really bored.

As to the Venn diagrams, I have a two-ring and a three-ring template
. . already designed and stored in my archives.