Parallel inequalities have the form:
ax+by+p<0, ax+by+q≤0, ax+by+r≥0, ax+by+s>0, where a and b are constants, and p, q, r and s are different constants. Graphically, less than (<) means the area to the left of the line excluding points on the line, less than or equal (≤) is the left area including line points, greater than (>) is the right area excluding line points, and greater than or equal (≥) is the right area including line points.
If two inequalities are graphed we can have the area between the lines, when the area to the right of one line coincides with the area to the left of the other. If the line points are excluded in each case one inequality will be > while the other is <.
When one inequality is ≥ points on the upper line are included while the lower line is < and excludes line points.
One line with > and the other with ≤ means exclude the line points on the upper line but include them on the lower one.
If upper line is ≥ and the lower is ≤ then all the line points are included.
If the upper line is < or ≤ and the lower is > or ≥ there is no common area and the area between the lines is excluded.
If both inequalities are similar (both are < or ≤; or both are > or ≥) then one inequality is absorbed by the other because the area bounded by one includes the area bounded by the other.
The four different graphs would be:
(1) Both areas are to the left of one of the inequalities (absorption)
(2) Both areas are to the right of one of the inequalities (absorption)
(3) There is a common area between the inequalities where both inequalities are satisfied. The directions of the inequality signs are opposite.
(4) There's an exclusion area between the inequalities where neither inequality is satisfied. The directions of the inequality signs are opposite.