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Chapter P
Prerequisites:
Fundamental
Concepts of
Algebra 1
P.9 Linear Inequalities
and Absolute Value
Inequalities
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Objectives:
•
•
•
•
•
Use interval notation.
Find intersections and unions of intervals.
Solve linear inequalities.
Solve compound inequalities.
Solve absolute value inequalities.
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Solving an Inequality
Solving an inequality is the process of finding the set
of numbers that make the inequality a true statement.
These numbers are called the solutions of the inequality
and we say that they satisfy the inequality. The set of
all solutions is called the solution set of the inequality.
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Interval Notation
The open interval (a,b) represents the set of real
numbers between, but not including, a and b.
(a, b)   x a  x  b
The closed interval [a,b] represents the set of real
numbers between, and including, a and b.
[a, b]   x a  x  b
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A half-open, or half-closed interval is
(a, b], consisting of all real numbers x
for which a < x < b.
(
a
]
b
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A half-open, or half-closed interval is
[a, b), consisting of all real numbers x
for which a < x < b.
[
a
)
b
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[
a
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(
a
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Interval Notation (continued)
The infinite interval (a, ) represents the set of real
numbers that are greater than a.
(a, )   x x  a
The infinite interval ( , b] represents the set of real
numbers that are less than or equal to b.
(, b]   x x  b
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Parentheses and Brackets in Interval Notation
Parentheses indicate endpoints that are not included in
an interval. Square brackets indicate endpoints that are
included in an interval. Parentheses are always used
with  or  .
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Example: Using Interval Notation
Express the interval in set-builder notation and graph:
[1, 3.5]
x 1  x  3.5
Express the interval in set-builder notation and graph:
(, 1)
x x  1
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Write the inequality -3 < x < 2 using
interval notation. Illustrate the inequality
using a real number line.
 3,2
[
-3
0
)
2
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Sets and Intervals
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Sets and Intervals
A set is a collection of objects, and these objects are
called the elements of the set. If S is a set, the notation a
 S means that a is an element of S, and b  S means
that b is not an element of S.
For example, if Z represents the set of integers, then
–3  Z but   Z.
Some sets can be described by listing their elements
within braces. For instance, the set A that consists of all
positive integers less than 7 can be written as
A = {1, 2, 3, 4, 5, 6}
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Sets and Intervals
We could also write A in set-builder notation as
A = {x | x is an integer and 0 < x < 7}
which is read “A is the set of all x such that x is an
integer and 0 < x < 7.”
If S and T are sets, then their union S  T is the set that
consists of all elements that are in S or T (or in both).
The intersection of S and T is the set S  T consisting
of all elements that are in both S and T.
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Sets and Intervals
In other words, S  T is the common part of S and T.
The empty set, denoted by Ø, is the set that contains no
element.
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Example 4 – Union and Intersection of Sets
If S = {1, 2, 3, 4, 5}, T = {4, 5, 6, 7}, and V = {6, 7, 8},
find the sets S  T, S  T, and S  V.
Solution:
S  T = {1, 2, 3, 4, 5, 6, 7}
S  T = {4, 5}
SV=Ø
All elements in S or T
Elements common to both
S and T
S and V have no element
in common
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Finding Intersections and Unions of Two Intervals
1. Graph each interval on a number line.
2. a. To find the intersection, take the portion of the
number line that the two graphs have in common.
b. To find the union, take the portion of the number
line representing the total collection of numbers
in the two graphs.
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Example: Finding Intersections and Unions of Intervals
Use graphs to find the set: [1,3]  (2,6).
Graph of [1,3]:
Graph of (2,6):
Numbers in both [1,3] and (2,6):
Thus,
[1,3]  (2,6)  (2,3].
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Solving Linear Inequalities in One Variable
A linear inequality in x can be written in one of the
following forms  a  0  :
ax  b  0
ax  b  0
ax  b  0
ax  b  0
In general, when we multiply or divide both sides of
an inequality by a negative number, the direction of
the inequality symbol is reversed.
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Example: Solving a Linear Inequality
Solve and graph the solution set on a number line:
2  3x  5.
2  3x  5
3x  3
3x 3

3 3
x  1
The solution set is
x x  1 .
The interval notation for this
solution set is [1, ) .
The number line graph is:
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Example: Solving a Compound Inequality
Solve and graph the solution set on a number line:
1  2 x  3  11.
Our goal is to isolate x in the middle.
1  2 x  3  11
In interval notation,
2  2 x  8
the solution is [-1,4).
1  x  4
In set-builder notation, the solution set is
x 1  x  4
The number line graph looks like
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Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive
number,
1. The solutions of u  c are the numbers that satisfy
c  u  c.
2. The solutions of u  c are the numbers that satisfy
u  c
u  c.
or
These rules are valid if  is replaced by  and
 is replaced by  .
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Example: Solving an Absolute Value Inequality
Solve and graph the solution set on a number line:
18  6  3x .
We begin by expressing the inequality with the
absolute value expression on the left side:
6  3x  18
We rewrite the inequality without absolute value bars.
6  3x  18 means 6  3 x  18 or 6  3x  18.
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Example: Solving an Absolute Value Inequality
(continued)
We solve these inequalities separately:
6  3 x  18
6  3 x  18
3 x  24
3 x  12
3x 12
3x 24


3
3
3 3
x  4
x 8
The solution set is
x x  4 or x  8
The number line graph looks like
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