Intermediate Algebra (11th edition)

An inequalitysays that two expressions are notequal. Solving inequalities is

similar to solving equations.

SECTION 2.5 Linear Inequalities in One Variable 91

OBJECTIVES

Linear Inequalities in One Variable

2.5

1 Solve linear
inequalities by
using the addition
property.

2 Solve linear
inequalities by
using the
multiplication
property.

3 Solve linear
inequalities with
three parts.

4 Solve applied
problems by using
linear inequalities.

In Section 1.1,we used interval notation to write solution sets of inequalities.

● A parenthesis indicates that an endpoint is notincluded.

● A square bracket indicates that an endpoint is included.

We summarize the various types of intervals here.

NOTEA parenthesis is alwaysused next to an infinity symbol, -qor. q

Type of Set-Builder Interval Interval Notation Notation Graph Open interval

5 x|a 6 x 6 b 6 1 a, b 2

Half-open (or half-closed) interval

5 x|a 6 x...b 6 1 a, b 4

5 x|a...x 6 b 6 3 a, b 2 a b

a b

Disjoint interval*

5 x|x 6 a or x 7 b 6 1 - q, a 2 ́ 1 b, q 2

Closed interval

5 x|a...x...b 6 3 a, b 4

Infinite interval

is a real number 0

5 x|x 6 1 - q, q 2

a

5 x|x...a 6 1 - q, a 4

a

5 x|x 6 a 6 1 - q, a 2

a

5 x|xÚa 6 3 a, q 2

a

5 x|x 7 a 6 1 a, q 2

*We will work with disjoint intervals in Section 2.6when we study set operationsand compound inequalities.

Linear Inequality in One Variable

A linear inequality in one variablecan be written in the form

or ,

where A, B, and Care real numbers, with AZ0.

AxB<C, AxB◊ C, AxB>C, AxB»C

x+ 56 2, x- 3 Ú5, and 2 k+ 5 ... 10 Examples of linear inequalities