Section 5.5Systems 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

x

y

Example

Graph: y 3x-4

x

y

Graphing Linear Inequalities without Using Test Points

Example

x

y

Graph the inequality on the graph below:

x -2

Example

x

y

Graph the inequality on the graph below:

y 1

Graphing a Nonlinear Inequality in Two Variables

Example

2 2Graph x 16y

x

y

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

2

Graph the solution set of the system:

y< -x +4

y -3x+2

x

y

Applications

ExampleMany 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.

(a)

(b)

(c)

(d)

3 2

4

3

3 2

y x

y x

y x

y x

Which equation describes the graph above.

x

y

(a)

(b)

(c)

(d)

The intersection of two regions results in the green region

at the top of the parabola. Which two inequalities would give

this intersection?2

2

2

2

3; 1

3; 1

3; 1

3; 1

y x y x

y x y x

y x y x

y x y x