Download Graphs of Inequalities Solving Inequalities Using the Addition principle

Math 52
2.7 "Solving Inequalities"
Objectives:
*
Determine whether a given number is a solution of an inequality.
*
Graph an inequality on a number line.
*
Solve inequalities using the addition and multiplication principles.
Preliminaries:
In order to solve inequalities we need to remember the de…nition of an inequality, the inequality symbols covered in
Section 1.2, and the idea of solving an inequality.
De…nition:
"Inequality"
An inequality is a number sentence with >; <;
For example,
Inequality Symbols:
<
"is less than"
"is less than or equal to"
>
or
as its verb.
"is greater than"
"is greater than or equal to"
Solution:
A replacement that makes an inequality true is called a solution. The set of all solutions is called the solution set.
When we have found the set of all solutions of an inequality, we say that we have solved the inequality.
Graphs of Inequalities
A graph of an inequality is a drawing that represents its solutions. An inequality in one variable can be graphed on a
number line.
Example 1: (Graphing inequalities)
Graph the following inequalities.
a) x < 2
b) x
3
Solving Inequalities Using the Addition principle
Now we extend our equation-solving principles to the solving of inequalities. First we need to go over the idea of
set-builder notation which we will use to represent the solution of inequalities.
Set-builder Notation
Set-builder Notation
Graph
Page: 1
Graph
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
2.7
Addition and Subtraction Properties of Inequalities:
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
Example 2: (Solving inequalities by using the addition and subtraction principles)
Solve and graph the following inequalities.
1
1
a) x 3
10
b) 3x + > 2x +
8
2
Solving Inequalities Using the Multiplication Principle
Multiplication and Division Properties of Inequalities:
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 3: (Solving inequalities by using the multiplication and division properties)
Solve and graph the following inequalities.
6
a) 2y 12
b)
4x
5
Page: 2
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
2.7
Using the Principles Together
All of the equation-solving techniques used in Sections 2:1 2:3 can be used with inequalities, provided we remember to
reverse the inequality symbol when multiplying or dividing on both sides by a negative number.
Example 4: (Solving inequalities using the principles together)
Solve and graph the following inequalities.
a) 5x 9 21
b) 18
c) 2 (3 + 4m)
9
e) 2:1x + 45:2 > 3:2
45
6y
d) 3 (r
8:4x
f) 1
Page: 3
4y < 63 + 5y
6) + 2 < 4 (r + 2)
2
y
3
9
5
21
y 3
+
5 5
Notes by Bibiana Lopez