Question

How do you graph the inequality `-9<=2x+y`?

Answer 1

Solve for `y` and shade the side with points that make the inequality true.

Explanation:

First, solve for `y` as if you had the equality

`-9<=2x+y`

Solving for `y` gives

`-9color(red)(-2x)<=2xcolor(red)(-2x)+y`

`-9-2x<=y`

We usually look at this with the `y` on the left hand side

`y>=-9-2x`

The inequality symbol is `>=`, meaning greater than or equal to, so draw this as a solid line on your graph. You can find two values by plotting points.

graph{y=-9-2x [-10, 5, -15, 5]}

Finally, you can see that the point `(0,0)` is on the right side of the line and the point `(-5,0)` is on the left side of the line.

To determine which side to shade, plug in these values and find which one makes the inequality true and which one makes the inequality false.

For `(0,0)`, we get

`y>=-9-2x`
`0>=-9-2(0)`
`0>=-9` which is true. Zero is bigger (or equal to) `-9`

For `(-5,0)`, we get

`y>=-9-2x`
`0>=-9-2(-5)`
`0>=-9+10`
`0>=1` which is false. Zero is not bigger than `1`.

Thus, we shade on the side containing the point `(0,0)`

graph{y>=-9-2x [-10, 5, -15, 5]}