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Math Help - Inequality

  1. #1
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    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...
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    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}

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