1. ## Inequality

Hi! I really forgot how I have to solve this equations. Could someone give me a hint?

Thank you very much!

3x+4<12x-5<1 =

and then:

Write down the inequalities that define the interior of a triangle with vertices (-1, -1), (3, 1) and (1,5)

I am so confused...

2. Hello, James!

Find the inequalities that define the interior of a triangle
with vertices: . $A(\text{-}1, \text{-}1),\;B(3, 1),\;C(1,5)$
First, graph the points . . .
Code:
              |   C
|   o (1,5)
|  *  *
| *     *
|*        * B
*           o (3,1)
*|       *
------*-+---*------------
*  *
A o   |
(-1,-1)|
|

Find the equation of line $AB.$

. . Slope of $AB\!:\;\;m \:=\:\frac{1-(\text{-}1)}{3-(\text{-}1)} \:=\:\frac{1}{2}$

. . Equation: . $y - 1 \:=\:\tfrac{1}{2}(x-3) \quad\Rightarrow\quad y \:=\:\tfrac{1}{2}x - \tfrac{1}{2}$

. . We want the region above this line: . $y \:> \:\tfrac{1}{2}x - \tfrac{1}{2}$

Find the equation of line $BC.$

. . Slope of $BC\!:\;\;m \:=\:\frac{5-1}{1-3} \:=\:-2$

. . Equation: . $y - 5 \:=\:-2(x-1) \quad\Rightarrow\quad y \:=\:\text{-}2x + 2$

. . We want the region below this line: . $y \:<\:\text{-}2x + 7$

Find the equation of line $AC.$

. . Slope of $AC\!:\;\;m \:=\:\frac{5-(\text{-}1)}{1 - (\text{-}1)} \:=\:3$

. . Equation: . $y - 5 \:=\:3(x-1) \quad\Rightarrow\quad y \:=\:3x+2$

. . We want the region below this line: . $y \:<\:3x+2$

Therefore: . $\begin{Bmatrix} y &>& \frac{1}{2}x - \frac{1}{2} \\ \\[-4mm] y &<& \text{-}2x + 7 \\ \\[-4mm] y &<& 3x + 2 \end{Bmatrix}$