CHAPTER 8
More on Inequalities
Source:
Elementary & Intermediate Algebra, Fourth Edition
Prepared and Edited by:
Dr. Mohamad Hammoudi

Topics of Chapter 8
8.2: Solving Absolute-Value Equations and
Inequalities in One Variable
8.3: Solving Systems of Linear Inequalities in Two
Variables
2

8.2: Solving Absolute-Value Equations
Objectives:
8.2.1: Find the absolute value of an expression
8.2.2: Solve an absolute-value equation
3

8.2.1: Find the Absolute Value of an Expression
The
absolute value
of a real number is the distance from
that real number to 0. Because absolute value is a distance,
it is always
positive
. Formally, we say
Examples:
Find the absolute value for each expression.
(a)
ǀ
–
3
ǀ
(b)
ǀ
7
–
2
ǀ
(c)
ǀ
–
7
–
2
ǀ
4

8.2.2: Solve an Absolute-Value Equation
Given an equation such as
ǀ
x
ǀ
= 5, there are two possible
solutions. The value of
x
could be 5 or
–
5. In either case,
the absolute value is 5. This can be generalized in the
following property of absolute-value equations.
The variable must be positive because an equation such as
ǀ
x
ǀ
=
–
5 has no solution.
5

8.2.2: Solve an Absolute-Value Equation
For
ǀ
x
ǀ
= +
p,
where p>0,
x = p or
x =
–
p
S.S. = {p, -p}
For
ǀ
x
ǀ
= 0
x = 0
S.S. = {0}
For
ǀ
x
ǀ
= -
p,
where p<0,
No solution
S.S. = { } or
Ø
6

8.2.2: Solve an Absolute-Value Equation

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