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Hawkes Learning Systems:
College Algebra
Section 2.2: Solving Linear Inequalities
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Objectives
o Solving linear inequalities.
o Solving compound linear inequalities.
o Solving absolute values inequalities.
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Solving Linear Inequalities
o If the equality symbol in a linear equation is replaced
with  ,  , , or  , the result is a linear inequality.
o For example, a linear inequality in x is of the form
ax  b  c where b and c are real numbers and a  0.
o The solution of a linear inequality typically consists
of some interval of real numbers described in set
notation, graphically or with interval notation.
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Solving Linear Inequalities
Cancellation Properties for Inequalities
Throughout this table, A, B and C, represent algebraic expressions. These properties
are true for all inequalities.
Property
A B  AC  B C
Description
Adding the same quantity to both sides of
an inequality results in an equivalent
inequality.
If C  0, A  B  A  C  B  C
If both sides of an inequality are
multiplied by a positive quantity, the
sense of the inequality is unchanged.
If C  0, A  B  A  C  B  C
If both sides of an inequality are
multiplied by a negative quantity, the
sense of the inequality is reversed.
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Example: Linear Inequalities
Solve the following linear inequality.
10  4( x  2)  (2  x)
Step 1: Distribute.
Step 2: Combine like terms.
Step 3: Divide by 5 . Note
the reversal of the
inequality sign.
10  4 x  8  2  x
4 x  18  2  x
5 x  20
x4
Solution is  4, 
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Example: Linear Inequalities
Solve the following linear inequality.
7  v  5  9v

2
4
 7  v  5    9v 
2 4
   4
2

  4
14(v  5)  9v
14v  70  9v
5v  70
Divide by 5.
v  14
Inequality stays the same.
Solution is (,14)
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Graphing a Solution
o The solutions in the previous examples were
described using interval notation, but graphing can
also be used to describe solutions.
o Like in interval notation, parentheses are used when
endpoints are not included in the interval and
brackets are used when the endpoints are included
in the interval.
o For example,

 4,  is graphed as follows:
4

 ,14 is graphed as follows:
14
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Solving Compound Linear Inequalities
A compound inequality is a statement containing two
inequality symbols, and can be interpreted as two
distinct inequalities joined by the word “and”.
For example, in a course where the grade depends
solely on the grades of 5 exams, the following
compound inequality could be used to determine the
final exam grade needed to score a B in the course.
92  65  71  80  x
80 
 90
5
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Example: Solving Compound Inequalities
Solve the compound inequality from the previous slide.
92  65  71  80  x
80 
 90
5
400  308  x  450
Step 1: Multiply all
sides by 5.
Step 2: Subtract 308
from all sides.
92  x  142
Solution is 92,142 
Note: If this compound inequality relates to test scores, as
indicated on the previous slide, the solution set is 92,100,
assuming 100 is the highest score possible.
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Example: Solving Compound Inequalities
Solve the compound inequality.
12  5  2  x   15
12  10  5 x  15
Note: each inequality
is reversed since we
are dividing by a
negative number!
2  5 x  25
2
 x  5
5
2
5  x 
5
 2
Solution is  5, 
 5
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Solving Absolute Value Inequalities
An absolute value inequality is an inequality in which
some variable expression appears inside absolute value
symbols.
x can be interpreted as the distance between x and
zero on the real number line. This means that absolute
value inequalities can be written without absolute
values as follows, assuming a is a positive real number:
x  a  a  x  a
and
x  a  x  a or x  a
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Example: Solving Absolute Value Inequalities
Solve the following absolute value inequality.
5y  3  2  9
Step 1: Subtract 2.
Step 2: Rewrite the
inequality without
absolute values.
Step 3: Solve as compound
inequality.
5y  3  7
7  5 y  3  7
4  5 y  10
4
 y2
5
 4 
Solution is  , 2 
5 
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Example: Solving Absolute Value Inequalities
Solve the following absolute value inequality and graph
the solution.
6  2x  8
6  2 x  8
2 x  14
x7
or
6  2x  8
or
2 x  2
x  1
or
Solution is  , 1   7,  

1

7
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Example: Solving Absolute Value Inequalities
Solve the following absolute value inequality
10  x  7  4
x  7  6
Solution is 
The solution set is the empty set, as it is impossible for
the absolute value of any expression to be negative.
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Example: Solving Absolute Value
Inequalities
Solve the following absolute value inequality.
|2x – 1| < x + 4
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Translating Inequality Phrases
o Many real-world applications leading to inequalities
involve notions such as “is not greater than”, “at least
as great as”, “does not exceed”, and so on.
o Phrases such as these all have precise mathematical
translations that use one of the four inequality
symbols, , , ,  .
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Translating Inequality Phrases
Sign

Phrase
Example
“is greater than”
The average temperature in New
Mexico during the month of May is
greater than 75.
x  75

“at least as great as”
The average price for a large screen
digital TV is at least $900.
x > 900

“is less than”
The number of skiers on the mountain is
less than 100.

“is not greater than”
x  100
In a quality test, the number of
defective products cannot be greater
than 5.
x5