Thinking Mathematically by Robert Blitzer

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Thinking Mathematically by Robert Blitzer Powered By Docstoc
					1.7 Linear Inequalities
          and
   Absolute Value
    Inequalities
 (Rvw.) Graphs of Inequalities; Interval
              Notation
There are infinitely many solutions to the
 inequality x > -4, namely all real numbers
 that are greater than -4. Although we
 cannot list all the solutions, we can make a
 drawing on a number line that represents
 these solutions. Such a drawing is called the
 graph of the inequality.
    Graphs of Inequalities; Interval Notation
•       Graphs of solutions to linear inequalities are
    shown on a number line by shading all points
    representing numbers that are solutions.
    Parentheses indicate endpoints that are not
    solutions. Square brackets indicate endpoints
    that are solutions.
(baby face & block headed old man drawings) (Also see p 165 for more clarification.)
Do p 176 #110. Emphasize set builder, interval, and graphical solutions.
                     Text Example
Graph the solutions of
a. x < 3 b. x  -1          c. -1< x  3.

Solution:
a. The solutions of x < 3 are all real numbers that are
    _________ than 3. They are graphed on a number
    line by shading all points to the _______ of 3. The
    parenthesis at 3 indicates that 3 is NOT a solution,
    but numbers such as 2.9999 and 2.6 are. The arrow
    shows that the graph extends indefinitely to the
    _________.

   -5    -4     -3    -2    -1      0     1      2     3
Note: If the variable is on the left, the inequality symbol shows
the shape of the end of the arrow in the graph.
                Text Example
Graph the solutions of
a. x < 3 b. x  -1 c. -1< x  3.
Solution:
b. The solutions of x  -1 are all real numbers that are
    _________ than or ___________ -1. We shade all
    points to the ________ of -1 and the point for -1
    itself. The __________ at -1 shows that 1ISa
    solution for the given inequality. The arrow shows
    that the graph extends indefinitely to the________.

    -5    -4   -3    -2   -1     0    1    2     3
     Ex. Con’t. Graph the solutions of c. -1< x  3.
     Solution:
     c. The inequality -1< x  3 is read "-1 is ______ than
         x and x is less than or equal to 3," or "x is
         _________ than -1 and less than or equal to 3."
         The solutions of -1< x  3 are all real numbers
         between -1 and 3, not including -1 but including
         3. The parenthesis at -1 indicates that -1 is not a
         solution. By contrast, the bracket at 3 shows that 3
         is a solution. Shading indicates the other solutions.


         -5    -4     -3    -2     -1     0      1     2     3
Note: it must make sense in the original inequality if you take out
variable. In this case, does -1 < 3 make sense? If not, no solution.
                (Rvw.) Properties of Inequalities
   Property                        The Property In Words                  Example
   Addition and Subtraction        If the same quantity is added to or    2x + 3 < 7
   properties                      subtracted from both sides of an       subtract 3:
   If a < b, then a + c < b + c.   inequality, the resulting inequality   2x + 3 - 3 < 7 - 3
   If a < b, then a - c < b - c.   is equivalent to the original one.     Simplify: 2x < 4.

   Positive Multiplication         If we multiply or divide both sides    2x < 4
   and Division Properties         of an inequality by the same           Divide by 2:
   If a < b and c is positive,     positive quantity, the resulting       2x  2 < 4  2
   then ac < bc.                   inequality is equivalent to the        Simplify: x < 2
   If a < b and c is positive,     original one.
   then a  c < b  c.

   Negative Multiplication         if we multiply or divide both sides    -4x < 20
   and Division Properties         of an inequality by the same           Divide by –4 and
   If a < b and c is negative,     negative quantity and reverse the      reverse the sense of
   then ac  bc.                   direction of the inequality symbol,    the inequality:
   If a < b and c is negative,     the result is an equivalent            -4x  -4  20  -4
   then a  c  b  c.             inequality.                            Simplify: x  -5


Bottom line: treat just like a linear EQUALITY, EXCEPT you flip
the inequality sign if:
    Ex: Solve and graph the solution set on a number
               line: 4x + 5  9x - 10.
Solution We will collect variable terms on the left and constant terms on
the right.
    4x + 5  9x - 10               This is the given inequality.




The solution set consists of all real numbers that are _________ than or
equal to _____, expressed in interval notation as ___________. The graph
of the solution set is shown as follows:


Do p 175#58, 122
        (Rvw) Solving an Absolute
            Value Inequality
 If X is an algebraic expression and c is a positive
     number:
 1. The solutions of |X| < c are the numbers that
     satisfy -c < X < c. (less thAND)
 2. The solutions of |X| > c are the numbers that
     satisfy X < -c or X > c. (greatOR) To put
     together two pieces using interval notation, use
     the symbol for “union”:
 These rules are valid if < is replaced by  and > is
    replaced by .
*** IMPORTANT: You MUST ISOLATE the absolute value before
applying these principles and dropping the bars.
     Text Example (Don’t look at notes, no need to write.)
 Solve and graph: |x - 4| < 3.
Solution      |X| < c means -c < X < c

              |x - 4| < 3 means -3< x - 4< 3
We solve the compound inequality by adding 4 to all three
parts.
                          -3 < x - 4 < 3
                      -3 + 4 < x - 4 + 4 < 3 + 4
                            1< x<7
The solution set is all real numbers greater than 1 and less
than 7, denoted by {x| 1 < x < 7} or (1, 7). The graph of
the solution set is shown as follows:

(do p 175 # 72, |x + 1| < -2, | x + 1 | > -2)

				
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