Download Read and Interpret Inequality Symbols Unbounded Intervals

Math 35
4.1 "Solving Linear Inequalities in One Variable"
Objectives:
* Read and interpret inequality symbols.
* Graph intervals and use interval and set-builder notation.
* Solve linear inequalities using properties of inequality.
Read and Interpret Inequality Symbols
Inequalities are statements indicating that two expressions are unequal and they contain one or more of the following
symbols.
Inequality Symbol
Meaning
Example
Inequality Symbol
is not equal to
Meaning
Example
is less than or equal to
is less than
is greater than or equal to
is greater than
Note:
> By de…nition, a
> Similarly, a
b is true if either a < b or a = b is true.
b is true if either a > b or a = b is true.
De…nition:
"Linear Inequalities"
A linear inequality in one variable (say, x) is any inequality that can be written in one of the following forms,
where a; b; and c represent real numbers and a 6= 0:
Examples
:
Non-examples
Unbounded Intervals
Set-builder
Interval
Notation
Notation
Graph
Page: 1
Set-builder
Interval
Notation
Notation
Graph
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
4.1
Solve Linear Inequalities Using Properties of Inequality
To solve a linear inequality means to …nd the values of its variable that make the inequality true. The set of all solutions
of an inequality is called its solution set. We will use the following properties to solve inequalities in one variable.
Addition and Subtraction Properties of Inequality:
Adding or subtracting the same number from both sides of an inequality does not change the solutions. For any real
numbers a; b; and c :
and
Similar statements can be made for the symbols
, > , or
Multiplication and Division Properties of Inequality:
Multiplying or dividing by the same positive number does not change the solutions. For any real numbers
a; b; and c (where c is positive) :
If we multiply or divide by a negative number, the direction of the inequality must be reversed. For any
real numbers a; b; and c (where c is negative) :
Similar statements can be made for the symbols
; >; or
Example 1: (Solving inequalities)
Solve the following inequalities. Graph the solution set and write it using interval notation.
a) 8x + 4 <
44
b) x + 4
Page: 2
5x
1 + 2x
15
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
c)
2
(x + 2)
3
3
(x
5
3)
4.1
d) 3 (x
2) > 2 (x + 7)
Example 2: (Solving inequalities)
Solve the following inequalities. Graph the solution set and write it using interval notation.
4d 5
2 (2d 3)
a) 8n + 10 1 2 (4n 2)
b)
>
10
10
Page: 3
Notes by Bibiana Lopez