Graphs of Linear Inequalities

When the equal sign in a linear equation is replaced with an inequality sign, a linear inequality is formed. Solutions of linear inequalities are ordered pairs.

Example

Solution

Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5.

The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

3(1) – 5 5

–2 < 5 TRUE

3x – y < 5

3(6) – (–2) 5

20 < 5 FALSE

3x – y < 5

Example

Solution

Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5.

The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

3(1) – 5 5

–2 < 5 TRUE

3x – y < 5

3(6) – (–2) 5

20 < 5 FALSE

3x – y < 5

The graph of a linear equation is a straight line. The graph of a linear inequality is a half-plane, with a boundary that is a straight line. To find the equation of the boundary line, we simply replace the inequality sign with an equals sign.

Example Graph .y xSolution

First: Graph the boundary

y = x.

Since the inequality is greater than or equal to, the line is drawn solid and is part of the graph of

.y x

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-3

2

-2

3

-1

1

6

54

y = x

-4

-5Second: We choose a test point on one side of the boundary, say (0, 1).

Substituting into the inequality we get

1 0 True

We finish drawing the solution set by shading the half-plane that includes (0, 1).

(0, 1)

Examples Graph the following inequalities.

a.) 3 8y x

b.) 3 4 12x y

c.) 5 1 16x

d.) 5 2 1 11x

Systems of Linear Inequalities

To graph a system of equations, we graph the individual equations and then find the intersection of the individual graphs. We do the same thing for a system of inequalities, that is, we graph each inequality and find the intersection of the individual graphs.

Example The graph of the system

2,

3,

,

x y

x

y x

Let’s look at 6 different types of problems that we have solved, along with illustrations of each type.

Type Example Solution

Linear equations 2x – 8 = 3(x + 5) A numberin one variable

Graph

Type Example Solution

Linear inequalities –3x + 5 > 2 A set of numbers;in one variable an interval

Graph

Type Example Solution

Linear equations 2x + y = 7 A set of orderedin two variables pairs; a line

Graph

Type Example Solution

Linear inequalities x + y ≥ 4 A set of orderedin two variables pairs; a half-plane

Graph

Type Example Solution

System of x + y = 3, An ordered pair orequations in 5x – y = –27 a (possibly empty)two variables set of ordered pairs

Graph

Type Example Solution

System of 6x – 2y ≤ 12, A set of ordered inequalities in y – 3 ≤ 0, pairs; a regiontwo variables y ≥ x of a plane

Graph