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what is 3x+y<-2 ?

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3 Answers

3x + y < -2
  • First, recognize that because it has an x and a y it is a linear equation (an equation for a line).
  • One of the formats for a linear equation is:
    • y = mx + b
      • m is the slope (the number next to the x, which for our equation is 3)
      • b is the y-intercept
  • So let's arrange our equation so it looks the same as the standard
    • 3x + y < -2 (I need to transfer that 3x to the other side... I can do that by subtracting it, but when I subtract from one side, I have to subtract the exact same thing from the other side to keep them even (balanced)
      • 3x + y - 3x < -2 - 3x
    • That cancels out the 3x on the left and leaves us with
      • y < -2 - 3x
    • Now we want to rearrange the right side of the equation, because it still doesn't look like our format, so we will move the -2 to the end. This leaves us with
      • y < -3x - 2
  • Now we need to graph our line, but we need some points. So pick a number for x, and it can be ANY number and we're going to say, "Ok, x is now 0."
    • Plug 0 into the spot where x is and solve the equation for y
      • y = -3x - 2 (for now we will pretend that it says = instead of <)
      • y = -3(0)- 2
      • y = 0 - 2
      • y = -2
      • So now I know that if x = 0, y = -2
      • The (x,y) coordinates for our answer are (0,-2)
  • Repeat that for another value of x... pick another number. So let's say x = 1
    • Plug 1 into the spot where x is and solve for y
      • y = -3x - 2
      • y = -3(1)- 2
      • y = -3 - 2
      • y = -5
      • So now I know that if x = 1, y = -5
      • The (x,y) coordinates for our answer are (1,-5)
  • Repeat that for another value of x... pick another number. So let's say x = -1
    • Plug -1 into the spot where x is and solve for y
      • y = -3x - 2
      • y = -3(-1)- 2
      • y = 3 - 2
      • y = 1
      • So now I know that if x = -1, y = 1
      • The (x,y) coordinates for our answer are (-1,1)
 
Now you have 3 points you can plot (0,-2), (1,-5), (-1,1). Graph the line!
 
And finally!!  Let's not ignore our < symbol any more... that tells me to shade in everything BELOW the line, because it's not actually EQUAL to the line, it's LESS THAN the line, so anything below the line solves the problem.
 
 You have to graph the line of  y = -3X -2
 
     Connect the intercepts:
 
        ( 0, -2 ) & ( -2/3, 0)
 
       Test a point like ( 0, 0) to the left of the line.
 
        Y < -2 - 3x
 
         0 < -2- 3(0)
 
       Therefore plane to the left of the line is the solution. Hash mark it as Solution.
Move 3x to the right. You will get 
y < -2-3x
Now, the solution of this inequality consists of all points of the coordinate plane that lie below the line given by the equation y=-3x-2