When do you change the sign for graphing inequalities I mean: 3 < -2y+6x, does the symbol's direction change?

Answer 1

3 < -2y + 6x

Let's try to get y by itself.

First, move the 6x over to the other side:
3 - 6x < -2y + 6x - 6x
3 - 6x < -2y + 0
3 - 6x < -2y

Next, we multiply both sides by (-1/2). Recall that multiplying or dividing by a negative numbers means that we have to switch the inequality sign:
(-1/2)(3-6x) > (-1/2)(-2)y
3x - (3/2) > y

So our graph is now:
y < 3x - (3/2)

To graph this, I always consider the equality to see what kind of graph it is. Then I would think about the inequality when I do my shading.

So, y = 3x - 3/2 is a straight line with slope 3 and y intercept at (0,-3/2).

Since there is no equal sign in our inequality (i.e it is < and not <=), we know that we will draw this line dotted to indicate that it is not part of our solution. In other words, no point that lies ON the line will satisfy y < 3x - (3/2).

As for shading, y < 3x - 3/2 gives us that the solutions lie BELOW the dotted line so you would shade below. To confirm this, pick a point below the line and a point above the line and plug in both points. Whichever point satisfies the inequality will give you a clue as to which area you will shade (i.e. above or below).

Intuition for switching inequality signs:
Pick any two numbers. Let's say 2 and 3 for example.
Clearly, 2 < 3.
If we multiply or divide both numbers by -1, we get -2 and -3.
Notice that -2 > -3

Answer 2

As you manipulate 3<-2y+6x
you will change the direction of the sign any time yo multiply or divide both sides by a negative number.
For example, you can add 2y to both sides, and you're fine:

2y+3 < 6x
And you can subtract 3 from both sides:
2y < 6x - 3
and divide by 2
y < 3x - 3/2
But you cannot multiply by -1:
-y < 3/2 - 3x would be incorrect. It needs to be -y > 3/2 - 3x

Answer 3

yes