Equality

[mathematics]

I am in need of this help pleaseee

Determine the solution region of ; 3x(2y-3)<0

Please show me how to do this

 

[soroban]

Hello, mathematics!

Determine the solution region of: .\(\displaystyle 3x(2y-3)\:<\:0\)

\(\displaystyle \text{The product of }3x\text{ and }2y-3\text{ is negative.}\)


\(\displaystyle \text{There are two cases:}\)

. . \(\displaystyle [1]\;3x \,>\,0\:\text{ and }\:2y-3\:<\:0\)
. . \(\displaystyle [2]\;3x\,<\,0\:\text{ and }\:2y-3\,>\,0\)


\(\displaystyle \text{Case [1]: }\;\begin{array}{ccccccc}3x \:>\: 0 & \Rightarrow & x \:>\: 0 \\2y-3 \:<\: 0 & \Rightarrow & y \: < \: \frac{3}{2} \end{array}\)


\(\displaystyle \text{Case [2]: }\;\begin{array}{cccc}3x\:<\:0 & \Rightarrow & x \:<\:0 \\ 2y-3 \:>\:0 & \Rightarrow & y \:>\:\frac{3}{2} \end{array}\)


\(\displaystyle \text{The solution region looks like this:}\)

           :::::|
          ::::::|
         :::::::|
        ::::::::|
      - - - - - + - - - - - -  y = 3/2
                |:::::::::
                |::::::::
      ----------+:-:-:-:-------
               0|::::::
                |:::::
                |::::