Question

How do you graph `y+4<=1/2(x-4)`?

Answer 1

See below.

Explanation:

Subtract 4 from both sides to get `y <= 1/2(x - 4) - 4`
Simplify right side to get `y <= 1/2x - 6`

Because this is also an equal to inequality we can say that

`y = 1/2x - 6`

This gives us the equation of a line that can be plotted.
All values that lie on the line are included values. The shaded area of included values can be found by taking a set coordinates of from either side of the line and checking them in the inequality to see if they satisfy it.

See graph:
graph{y + 4 <= (1/2)(x-4) [-25.67, 25.66, -12.83, 12.84]}

Answer 2

Please see below.

Explanation:

As `y+4 <= 1/2(x-4)`. we have

`y <= 1/2x-2-4` or `y <= 1/2x-6`

Hence first draw te graph of `y=1/2x-6`

When `x=0`, we have `y=-6` and when `y=0`, `x=12`

Hence we can draw the graph of `y=1/2x-6` by joining poiints `(0,-6)` and `(12,0)`. The graph appears as follows:

graph{(x-2y-12)(x^2+(y+6)^2-0.02)((x-12)^2+y^2-0.02)=0 [-4.46, 15.54, -7.28, 2.72]}

This divides the Cartesian plane in three parts,

One - on the line - all these points satisfy the inequality `y+4 <= 1/2(x-4)` as on the line `y+4=1/2(x-4)`. Observe that `y+4 <= 1/2(x-4)` has equality sign.

Two - to the left of line. Let us pick up the point `(0,0)` as for this `0+4 <= 1/2(0-4)` or `4 <= -2` and this does not satisfy `y+4 <= 1/2(x-4)`

Three - to the right of line. Let us pick up `(10,-5)` and at this we have `-5+4 <= 1/2(10-4)` or `-1<= 3` and this does satisfy `y+4 <= 1/2(x-4)`.

As the line and portion of the plane to the right of it satisfies the inequality `y+4 <= 1/2(x-4)`, the graph apears as follows:

graph{y+4 <= 1/2(x-4) [-4.46, 15.54, -7.28, 2.72]}

Note that if we had `y+4 < 1/2(x-4)`, only the third portion would have satisfied the equation. As points on line then do not satisfy, we draw it as dotted as shown below.

graph{y+4 < 1/2(x-4) [-4.46, 15.54, -7.28, 2.72]}