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Intermediate Algebra
Exam 3 Material
Inequalities and Absolute Value
Inequalities
• An equation is a comparison that says two algebraic
expressions are equal
• An inequality is a comparison between two or three
algebraic expressions using symbols for:
greater than:
greater than or equal to:
less than:
less than or equal to:
• Examples:


x 15  3x  3
1
 3  x  4  1
2


Two part inequality
Three part inequality
.
Inequalities
• There are lots of different types of
inequalities, and each is solved in a
special way
• Inequalities are called equivalent if they
have exactly the same solutions
• Equivalent inequalities are obtained by
using “properties of inequalities”
Properties of Inequalities
•
Adding or subtracting the same number to all parts of an
inequality gives an equivalent inequality with the same sense
(direction) of the inequality symbol
Add 3
x  3  2 is equivalent to : x  5
•
Multiplying or dividing all parts of an inequality by the same
POSITIVE number gives an equivalent inequality with the
same sense (direction) of the inequality symbol
Divide by 3
3x  6 is equivalent to : x  2
•
Multiplying or dividing all parts of an inequality by the same
NEGATIVE number and changing the sense (direction) of the
inequality symbol gives an equivalent inequality
Divide by - 2
 2 x  8 is equivalent to : x  4
Solutions to Inequalities
• Whereas solutions to equations are usually sets
of individual numbers, solutions to inequalities
are typically intervals of numbers
• Example:
Solution to x = 3 is {3}
Solution to x < 3 is every real number that is less
than three
• Solutions to inequalities may be expressed in:
– Standard Notation
– Graphical Notation
– Interval Notation
Two Part Linear Inequalities
• A two part linear inequality is one that
looks the same as a linear equation
except that an equal sign is replaced by
inequality symbol (greater than, greater
than or equal to, less than, or less than or
equal to)
• Example:
x 15  3x  3
Expressing Solutions to Two
Part Inequalities
• “Standard notation” - variable appears alone on left side of
inequality symbol, and a number appears alone on right side:
x2
• “Graphical notation” - solutions are shaded on a number line
using arrows to indicate all numbers to left or right of where shading
ends, and using a parenthesis to indicate that a number is not
included, and a square bracket to indicate that a number is
included
]2
• “Interval notation” - solutions are indicated by listing in order the
smallest and largest numbers that are in the solution interval,
separated by comma, enclosed within parenthesis and/or square
bracket. If there is no limit in the negative direction, “negative
infinity symbol” is used, and if there is no limit in the positive
direction, a “positive infinity symbol” is used. When infinity
symbols are used, they are always used with a parenthesis.
(, 2]
Solving
Two Part Linear Inequalities
• Solve exactly like linear equations
EXCEPT:
– Always isolate variable on left side of
inequality
– Correctly apply principles of inequalities
(In particular, always remember to reverse
sense of inequality when multiplying or
dividing by a negative)
Example of Solving Two Part
Linear Inequalities
x 15  3x  3
x 15  3x  9
 2x  6
x  3
When dividing by a negative, reverse sense of inequality !
Standard Notation Solution
3
]
(,  3]
Graphical Notation Solution
Interval Notation Solution
Three Part Linear Inequalities
• Consist of three algebraic expressions compared with
two inequality symbols
• Both inequality symbols MUST have the same sense
(point the same direction) AND must make a true
statement when the middle expression is ignored
• Good Example:
1
 3  x  4  1
2
• Not Legitimate:
1
 3  x  4  1 Inequality Symbols Don' t Have Same Sense
2
.
1
 3  x  4  1 - 3 is NOT  -1
2
Expressing Solutions to Three
Part Inequalities
• “Standard notation” - variable appears alone
in the middle part of the three expressions
being compared with two inequality symbols:
2 x 3
• “Graphical notation” – same as with two part
inequalities:
2
3
(
]
• “Interval notation” – same as with two part
inequalities:
(2, 3]
Solving
Three Part Linear Inequalities
• Solved exactly like two part linear
inequalities except that solution is
achieved when variable is isolated in the
middle
Example of Solving
Three Part Linear Inequalities
1
x  4  1
2
1
 3  x  2  1
2
3
 6  x  4  2
2 x  2
Standard Notation Solution
2
2
[
)
Graphical Notation Solution
[2, 2) Interval Notation Solution
Homework Problems
• Section: 2.8
• Page: 174
• Problems: Odd: 3 – 17, 21 – 25,
29 – 71
• MyMathLab Homework Assignment 2.8 for
practice
• MyMathLab Quiz 2.8 for grade
Sets
• A “set” is a collection of objects (elements)
• In mathematics we often deal with sets
whose elements are numbers
• Sets of numbers can be expressed in a
variety of ways:
A  3, 6, 7, 11 Four specific numbers in the set
B  x | x  4
C   3, 8
D   , 2
All numbers greater th an 4
All numbers  - 3 and  8
All numbers  2
Empty Set
• A set that contains no elements is called
the “empty set”
• The two traditional ways of indicating the
empty set are:


Intersection of Sets
• The intersection of two sets is a new set
that contains only those elements that are
found in both the first AND and second set
• The intersection of sets M and N is
indicated by M  N
• Given M  1, 2, 3, 4 and N  2, 4, 6, 8
M  N  2, 4
Union of Sets
• The union of two sets is a new set that
contains all those elements that are found
either in the first OR the second set
• The intersection of sets M and N is
indicated by M  N
• Given M  1, 2, 3, 4 and N  2, 4, 6, 8
M N 
1, 2, 3, 4, 6, 8
Intersection and Union Examples
• Given
C   3, 8 and D   , 2
• Find the intersection and then the union (it may
help to first graph each set on a number line)
• Find
CD
 3, 2
• Find
CD
 , 8
Compound Inequalities
• A compound inequality consists of two
inequalities joined by the word “AND” or by
the word “OR”
• Examples:
2 x  3  3 AND x  2  5
2  x  5 OR
x 3  5
Solving Compound Inequalities
Involving “AND”
• To solve a compound inequality that uses
the connective word “AND” we solve each
inequality separately and then intersect
the solution sets
2 x  3  3 AND x  2  5
• Example:
2 x  6 AND x  3
x  3 AND x  3
 ,  3 3, 
No number can be in both sets  
This means there is no solution
Solving Compound Inequalities
Involving “OR”
• To solve a compound inequality that uses
the connective word “OR” we solve each
inequality separately and then union the
solution sets
• Example:
2  x  5 OR x  3  5
 x  3 OR x  8
x  3 OR x  8
 3,   8, 
Always simplify   3, 
Solution
Homework Problems
• Section: 9.1
• Page: 626
• Problems: Odd: 7 – 61
• MyMathLab Homework Assignment 9.1 for
practice
• MyMathLab Quiz 9.1 for grade
Definition of Absolute Value
• “Absolute value” means “distance away from zero” on
a number line
• Distance is always positive or zero
• Absolute value is indicated by placing vertical parallel
bars on either side of a number or expression
Examples:
The distance away from zero of -3 is shown as:
3  3
The distance away from zero of 3 is shown as:
3
3
The distance away from zero of u is shown as:
u
Can' t be simplified , because value of " u" is unknown. However, its value is zero or positive.
Absolute Value Equation
• An equation that has a variable contained
within absolute value symbols
• Examples:
| 2x – 3 | + 6 = 11
| x – 8 | – | 7x + 4 | = 0
| 3x | + 4 = 0
Solving Absolute Value
Equations
• Isolate one absolute value that contains an
algebraic expression, | u |
– If the other side is negative there is no solution
(distance can’t be negative)
– If the other side is zero, then write:
• u = 0 and Solve
– If the other side is “positive n”, then write:
• u = n OR u = - n and Solve
– If the other side is another absolute value
expression, | v |, then write:
• u = v OR u = - v and Solve
Example of Solving
Absolute Value Equation
2 x  3  6  11
2x  3  5
2x  3  5 OR 2x  3  5
2 x  2
2x  8
x  1
x4
Example of Solving
Absolute Value Equation
x  9  7x  4  0
x  9  7x  4
x  9  7x  4
13  6 x
 13
x
6
OR
x  9  7 x  4
x  9  7 x  4
8x  5
5
x
8
Example of Solving
Absolute Value Equation
3x  6  2
3x  4
?
This says distance is negative - NOT POSSIBLE!
Equation has NO SOLUTION!

Absolute Value Inequality
• Looks like an absolute value equation
EXCEPT that an equal sign is replaced
by one of the inequality symbols
• Examples:
| 3x | – 6 > 0
| 2x – 1 | + 4 < 9
| 5x - 3 | < -7
Solving Absolute Value
Inequalities
1. Isolate the absolute value on the left side to
write the inequality in one of the forms:
| u | < n or | u | > n
2a. If | u | < n, then write and solve one of these:
u > -n AND u < n (Compound Inequality)
-n < u < n Preferred! (Three part inequality)
2b. If | u | > n, then write and solve:
u < -n OR u > n (Compound inequality)
3. Write answer in interval notation
Example: Solve: | 3x | – 6 > 0
1. Isolate the absolute value on the left side to
write the inequality in one of the forms:
| u | < n or | u | > n
3x  6
2a. If | u | < n, then write:
-n < u < n , and solve
2b. If | u | > n, then write:
u < -n OR u > n , and solve
Which form does this match?
3x  6 OR 3x  6
2b
Next?
Example Continued
3x  6 OR 3x  6
Solve each inequality separately :
x  2 OR
x2
3. Answer in interval notation :
(,  2)  (2, )
Example: Solve: | 2x -1 | + 4 < 9
1. Isolate the absolute value on the left side to
write the inequality in one of the forms:
| u | < n or | u | > n
2x 1  5
2a. If | u | < n, then write:
-n < u < n , and solve
2b. If | u | > n, then write:
u < -n or u > n , and solve
Which form does this match?
 5  2 x 1  5
2a
Next?
Example Continued
 5  2 x 1  5
 4  2x  6
2 x 3
How do we write the answer?
 2, 3
Absolute Value Inequality
with No Solution
• How can you tell immediately that the following
inequality has no solution?
5x  7  2
• It says that absolute value (or distance) is
negative – contrary to the definition of absolute
value
• Absolute value inequalities of this form always
have no solution:
u  n (where  n represents a negative number)

Does this have a solution?
2x  5  0
• At first glance, this is similar to the last example,
because “ < 0 “ means negative, and:
2 x  5 can' t be less than a negative number !
• However, notice the symbol is: 
• And it is possible that: 2 x  5  0
• We have previously learned to solve this as:
2x  5  0
2x  5
5
x
2
5
Solution is : x 
2
Solve this:
4x  5  0
• Remember that absolute value of a number is always
greater than or equal to zero, therefore the solution will
be:
• every real number except the one that makes this
absolute value equal to zero (the inequality symbol says
it must be greater than zero)
• Another way of saying this is that: The only bad value of
“x” is: 4 x  5  0
5
x
4
5 5 

• The solution, in interval notation is:   ,    ,  
4 4 

Homework Problems
• Section: 9.2
• Page: 635
• Problems: Odd: 1, 5 – 31, 35 – 95
• MyMathLab Homework Assignment 9.2 for
practice
• MyMathLab Quiz 9.2 for grade