Consider the straight line graph with equation y = x.
When x = 0, y = 0 and when x = 1, y = 1, and
so on. The line is a set of an infinite number of points. The
point A(2, 2) is a particular element of this set. The line
divides the Cartesian plane into two half-planes.

The set of points in the Cartesian plane which lie above the line form the upper half-plane, and the set of points which lie below the line
form the lower half-plane. B(2, 4) is in the upper
half-plane, and C(2, 1) is in the lower half-plane.

We can see from the graph that the y-coordinate of B is
greater than its x-coordinate.

The equation of the upper-half plane is y > x.

Similarly, the y-coordinate of C is less than its x-coordinate.

The equation of the lower-half plane is y < x.

Graphs of Half-Planes

Let us first consider the line with equation y = x + 4.

The graph of y = x + 4 divides the plane into two regions
(one above the line and the other below the line).

A test point is used to decide whether the solution of the
inequality is the region above or below the straight line.

If the straight line graph does not pass through the origin, then it is
convenient to take (0, 0) as the test point. Otherwise, we choose
any point that is not on the line.

The line y = x + 4 does not pass through the origin, so
we choose (0, 0) as the test point.

This is not true.

Upper region is represented by y > x + 4
Line is represented by y = x + 4
Lower region is represented by y < x + 4

Note:

The boundary line, y = x + 4, is included in the
required region and this is indicated by drawing it as a solid line.

Always indicate on the graph how you have shown the required region,
usually by shading the region that is rejected.

Example 13

Sketch the linear inequality y < x + 3 showing all
relevant points.

Solution:

Let y = x + 3

x-intercept:

y-intercept:

When x = 0, y = 3

Test point:

Note:

The boundary line, y = x + 3, is not included in the
required region and this is indicated by drawing it as an broken line.

The required region is unshaded as indicated on the graph.