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

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]}

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]}