College Algebra 9th txtbk

SECTION 7–7 Systems of Linear Inequalities in Two Variables 11
Z THEOREM 1 Graphs of Linear Inequalities in Two Variables
The graph of a linear inequality
Ax By C or Ax By C
with B 0, is either the upper half-plane or the lower half-plane (but not both)
determined by the line Ax By C.
If B 0, then the graph of
Ax C or Ax C
is either the left half-plane or the right half-plane (but not both) determined by
the line Ax C.
As a consequence of Theorem 1, we can build a simple and fast mechanical procedure
for graphing linear inequalities.
Z PROCEDURE FOR GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES
Step 1.
Step 2.
Step 3.
Graph Ax By C as a dashed line if equality is not included in the
original statement or as a solid line if equality is included.
Choose any point not on the line and substitute the coordinates into the
inequality. [It’s usually easiest to use the origin (0,0) if you’re sure that
it is not on the line.]
If the inequality is true when substituting in the test point, the graph of
the original inequality is the half-plane containing that point. If the
inequality is false, the graph of the original inequality is the half-plane
not containing that point.
EXAMPLE 1 Graphing a Linear Inequality
Graph: 3x 4y 12
5
y
SOLUTION
3x 4y 12
x
5
Step 1. Graph 3x 4y 12 as a solid line, since equality is included in the original
statement (Fig. 4).
Step 2. Pick a convenient test point above or below the line. The point (0, 0) will be
easy to test. Substituting (0, 0) into the inequality
3x 4y 12
3(0) 4(0) 0 12
5
Z Figure 4
produces a true statement; therefore, (0, 0) is in the solution set.
Step 3. The line 3x 4y 12 and the half-plane containing the origin form the graph
of 3x 4y 12 (Fig. 5).