Question

How do you graph the inequality `2(y-1) > 3(x+1)`?

Answer 1

See below:

Explanation:

One way to do it is to put the inequality into a form that is more recognizable as the graph of a line. For instance, I prefer to use the slope intercept form, so let's put this inequality into that form:

`2(y-1)>3(x+1)`

`2y-2>3x+3`

`2y>3x+5`

`y>3/2x+5/2`

If we ignore the inequality sign for a moment and graph the line `y=3/2x+5/2`, we get this:

graph{3/2x+5/2}

So now the question is to decide which side of the line we need to shade. We can do that by seeing whether the point `(0,0)` satisfies the inequality (we can use any point, but the origin is often quite easy to use):

`0>3/2(0)+5/2=>0>5/2color(white)(000)color(red)X`

And so we shade the side of the line that does not have the origin in it.

The line itself will be dotted to indicate that the points on the line do not satisfy the inequality:

graph{y-3/2x-5/2>0}