Section 7.5
Systems of Inequalities
Linear Inequalities in Two
Variables and Their Solutions
A solution of an inequality in two variables, x and y, is an
ordered pair of real numbers with the following property:
When the x-coordinate is substituted for x and the y-coordinate
is substituted for y in the inequality, we obtain a true statement.
Each ordered-pair solution is said to satisfy the inequality.
The Graph of a Linear Inequality
in Two Variables
A half-plane is the set of all points on one side of a line.
A half plane is the graph of an inequality that involves
> or <. The graph of an inequality that involves  or 
is a half-plane and a line. A solid line is used to show that
a line is part of a graph. A dashed line is used to show that
a line is not part of a graph.
Example
Graph: y  -2x+3
y




x













Example
Graph: y  3x-4
y




x













Graphing Linear Inequalities
without Using Test Points
Example
Graph the inequality on the graph below:
x  -2
y




x













Example
Graph the inequality on the graph below:
y 1
y




x













Graphing a Nonlinear Inequality
in Two Variables
Example
Graph x  y  16
2
2
y




x













Systems of Inequalities in
Two Variables
Two or more linear inequalities make up a system of linear
inequalities. A solution of a system of linear inequalities in
two variables is an ordered pair that satisfies each inequality
in the system. The set of all such ordered pairs is the solution
set of the system. Graph a system of inequalities in two variables
by graphing each inequality in the same rectangular coordinate
system. Then find the region, if there is one, that is common to
every graph in the system. This region of intersection gives a
picture of the system's solution set. See the steps below.
Example
Graph the solution set of the system:
y< -x 2 +4
y  -3x+2
y




x













Applications
Example
Many elevators have a capacity of 3000 lbs. If a child
weighs an average of 50 lbs and an adult weighs an
average of 175 lbs, write the inequality that describes when
x children and y adults will keep the elevator from being
overloaded. Graph the inequality, keeping x and y positive.

y







x









Which equation describes the graph above.
(a) y  3 x  2
(b) y  x  4
(c) y  x  3
(d) y  3 x  2
The intersection of two regions results in the green region
at the top of the parabola. Which two inequalities would give
this intersection?
(a) y   x 2  3; y  x  1
(b) y   x 2  3; y  x  1
(c) y   x 2  3; y  x  1
2
(d) y   x  3; y  x  1