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Math 35
4.3 "Solving Absolute Value Equations and Inequalities"
Objectives:
* Solve equations of the form jXj = k and with two absolute values.
* Solve inequalities of the form jXj < k and jXj > k:
Preliminaries:
Many quantities studied in mathematics, science, and engineering are expressed as positive numbers. To guarantee that
a quantity is positive, we often use absolute value. In this section, we will consider equations and inequalities involving the
absolute value of an algebraic expression. Examples:
Solve Equations of the Form X = k
Solving Absolute Value Equations:
To solve jXj = k, we need to solve the equivalent compound equation
;
where k is any positive number and X is any algebraic expression.
Example 1: (Solving absolute value equations)
Solve the following equations.
2
3
2
a)
x 2
=
5
5
5
WARNING!!!
b)
5
2
x+4 +1=1
3
When solving absolute value equations, isolate the absolute value before writing the equivalent compound
statement (Example 1).
Example 2: (Finding inputs of absolute value equations)
Let f (x) = j2x
3j. For what value(s) of x is f (x) = 7?
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
4.3
Solve Equations with Two Absolute Values
Solving Equations with Two Absolute Values:
For any algebraic expressions X and Y: To solve jXj = jY j , solve the compound equation
Example 3: (Solving equations with two absolute values)
Solve j2x
3j = j4x + 9j :
Solve Inequalities of the Form X < k
Solving jXj < k and jXj
k:
For any positive number k and any algebraic expression X:
> To solve jXj < k; solve the equivalent double inequality
:
> To solve jXj
:
k; solve the equivalent double inequality
Example 4: (Solve inequalities of the form jXj < k and jXj
k)
Solve the following inequalities and graph the solution sets.
a) j3x + 2j < 4
b) jx
Page: 2
8j
2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
4.3
Solve Inequalities of the Form X > k
Solving jXj > k and jXj
k:
For any positive number k and any algebraic expression X:
> To solve jXj > k; solve the equivalent compound inequality
:
> To solve jXj
:
k; solve the equivalent compound inequality
Example 5: (Solve inequalities of the form jXj
k and jXj > k)
Solve the following inequalities and graph the solution set.
2 x
a)
1
4
b)
3
x+2
4
1>3
The following summary shows how we can interpret absolute value in three ways assuming k > 0:
Graphic Description
Algebraic Description
jxj = k
jxj = k is equivalent to x = k or x =
jxj < k
jxj < k is equivalent to
jxj > k
jxj > k is equivalent to x > k or x <
Page: 3
k
k<x<k
k
Notes by Bibiana Lopez