Question

How do you graph the system of linear inequalities `x-y>7` and `2x+y<8`?

Answer 1

See below:

Explanation:

`x-y>7`
`2x+y<8`

Let's first graph the boundary lines for each graph, then figure out what needs to be shaded. To graph the boundary lines, I'll change the form of the equations to slope-intercept:

`x-y>7`

`-y> -x+7`

`y < x-7`

graph{x-7[-40,40,-20,20]}

To shade, does the origin fall within the solution?

`0<0-7=>0<-7color(white)(000)color(red)X`

And so we shade the other side:

graph{y -x+7< 0[-40,40,-20,20]}

`2x+y<8`

`y<2x+8`

graph{2x+8[-40,40,-20,20]}

Is the origin part of this solution?

`0<2(0)+8=>0<8color(white)(000)color(green)root`

So we shade that side:

graph{y-2x-8<0[-40,40,-20,20]}

To put the graphs together, you'll shade to the right of the line that is rightmost (so "above" the point of intersection, it's `x-y>7` and below the point of intersection it's `2x+y<8`).

The point of intersection sits at:

`x-7=y=2x+8`

`:. x-7=2x+8`

`x=-15`

`-15-7=y=-22`

`:. (-15,-22)`