# Thinking Mathematically by Robert Blitzer

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```					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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 views: 4 posted: 10/1/2012 language: English pages: 10