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Section 7.5

Systems of Inequalities

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Linear Inequalities in TwoVariables and Their Solutions

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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.

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The Graph of a Linear Inequalityin Two Variables

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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

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Example

Graph: y -2x+3

x

y

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Example

Graph: y 3x-4

x

y

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Graphing Linear Inequalitieswithout Using Test Points

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Example

x

y

Graph the inequality on the graph below:

x -2

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Example

x

y

Graph the inequality on the graph below:

y 1

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Graphing a Nonlinear Inequalityin Two Variables

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Example

2 2

Graph x 16y

x

y

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Systems of Inequalities inTwo Variables

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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 apicture of the system's solution set. See the steps below.

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Example

2Graph the solution set of the system:y< -x +4

y -3x+2

x

y

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Applications

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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.

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(a)(b)

(c)

(d)

3 24

3

3 2

y x

y x

y x

y x

Which equation describes the graph above.

x

y

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(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