Intermediate Algebra Unit 9 Inequalities and Absolute Value by gxdA9lh

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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       Two part inequality

  3  x  4  1
      1                    Three part inequality   .
      2
              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     When dividing by a negative, reversesense of inequality !

    x  3       Standard Notation Solution
             3
                 ]                Graphical Notation Solution


      (,  3]        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:
      3  x  4  1
           1
           2
• Not Legitimate:
      3  x  4  1 Inequality Symbols Don' t Have Same Sense
           1
           2                                              .
       3  x  4  1 - 3 is NOT  -1
            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

3
     1
       x  4  1
     2
      1
 3  x  2  1
      2
  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 numbersin the set
  B  x | x  4     All numbers greater than 4
  C   3, 8        All numbers - 3 and  8
  D   , 2        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   , 2, 3, 4 and N  2, 4, 6, 8
              1
  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   , 2, 3, 4 and N  2, 4, 6, 8
               1
  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
 • Example:      2 x  3  3 AND x  2  5
                     2 x  6 AND x  3
                       x  3 AND x  3
                         ,  3 3, 
No number can be in both sets 
                  This means thereis 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
     2x  3  6  11
         2x  3  5

2x  3  5 OR 2x  3  5
   2x  8        2x  2
    x4            x  1
         Example of Solving
       Absolute Value Equation

   x  9  7x  4  0
             x  9  7x  4

x  9  7x  4   OR     x  9  7 x  4
 13  6x                x  9  7 x  4
  13                         8x  5
      x
  6                                5
                               x
                                   8
       Example of Solving
     Absolute Value Equation

  3x  6  2
      3x  4    ?
This says distanceis 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 formdoes this match? 2b      Next?
    3x  6 OR 3x  6
         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 formdoes this match?    2a   Next?
     5  2 x 1  5
          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:
  2x  5 can't be less than a negative number!
• However, notice the symbol is: 
• And it is possible that: 2x  5  0
• We have previously learned to solve this as:
  2x  5  0
                                    5
  2x  5          Solution is : x 
  x
      5                             2
      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: 4x  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

								
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