Transcript Document

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
properties
If a < b, then a + c < b + c.
If a < b, then a - c < b - c.
If the same quantity is added to or
subtracted from both sides of an
inequality, the resulting inequality
is equivalent to the original one.
2x + 3 < 7
subtract 3:
2x + 3 - 3 < 7 - 3
Simplify: 2x < 4.
Positive Multiplication
and Division Properties
If a < b and c is positive,
then ac < bc.
If a < b and c is positive,
then a  c < b  c.
If we multiply or divide both sides
of an inequality by the same
positive quantity, the resulting
inequality is equivalent to the
original one.
2x < 4
Divide by 2:
2x  2 < 4  2
Simplify: x < 2
Negative Multiplication
and Division Properties
If a < b and c is negative,
then ac  bc.
If a < b and c is negative,
then a  c  b  c.
if we multiply or divide both sides
of an inequality by the same
negative quantity and reverse the
direction of the inequality symbol,
the result is an equivalent
inequality.
-4x < 20
Divide by –4 and
reverse the sense of
the inequality:
-4x  -4  20  -4
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)