How do you graph the compound inequality #3p+6<8-p# and #5p+8>=p+6#?

The first step is to bring both inequalities to a simplest possible form using invariant transformations (that is, those that produce equivalent inequalities).

#3p+6<8-p#
Add #p# to both sides and subtract #6# from both sides.
#4p<2#
Divide both sides by #4#.
#p<1/2# (simplified inequality 1)

#5p+8 >= p+6#
Subtract #p# and subtract #8# from both sides.
#4p>=-2#
Divide both sides by #4#.
#p >= -1/2# (simplified inequality 2)

Now it's easy to combine both simplified inequalities (1) and (2).
The first one restricts #p# from above to be less than #1/2#.
The second one restricts #p# from below to be greater or equal to #-1/2#.
Combining these restrictions, we come to an interval #p# is supposed to be in:
#-1/2 <= p < 1/2#

Graphically, it is represented by an interval on the X-axis #[-1/2,1/2)# where a square bracket on the left indicates that the left border point #p=-1/2# is included into an interval, while the parenthesis on the right indicates that the right border point #p=1/2# is not included into an interval.
Usually, an arrow on the side of strong inequality (#p=1/2#) might indicate that an end point is not included (not on this graph).
graph{sqrt(x+1/2)0+sqrt(1/2-x) 0 [-1, 1, -0.5, 0.5]}