Transcript Section 1.7 Linear Inequalities and Absolute Value Inequalities Interval Notation Example Express the interval in set builder notation and graph:  3, 2  0, 4  ,

Section 1.7
Linear Inequalities
and
Absolute Value Inequalities
Interval Notation
Example
Express the interval in set builder notation
and graph:
 3, 2
 0, 4
 , 2 
Intersections and
Unions of Intervals
Example
Find the set:
 2, 3
 0,4 
Example
Find the set:
 2, 3
 0,4 
Solving Linear Inequalities
in One Variable
A linear inequality in x can be written in one of the following
forms: ax+b<0, ax+b  0, ax+b>0 ax+b  0. In each form
a  0.
Example:
-x+7  0
-x  -7
x7
When we multiply or divide both sides of an inequality by
a negative number, the direction of the inequality symbol
is reversed.
Example
Solve and graph the solution set on a number line:
4x  5  x  7
Checking the solution of a linear inequality
on a Graphing Calculator
2x 1  x  4
Separate the inequality
into two equations.
y1  2 x  1
y2   x  4
The intersection of the two lines is at
(1,3). You can see this because both y
values are the same, – 3.
The region in the red box is where the
values of y1 is greater than y2.
Y2=-x+4
The region on the graph of the red box is
where y1 is greater than y2. This is
when x is greater than 1.
Y1=2x+1
Inequalities with
Unusual Solution Sets
The solution set could be the null set,. The solution
set could be all real numbers,  -,  .
x  x 1
0 1
Never true 
x  x 1
0 1
Always true  ,  
Example
Solve each inequality:
x 3 x  4
Solving
Compound Inequalities
Now consider two inequalities such as
-3<2x+1 and 2x+1  3
express as a compound inequality
-3<2x+1  3
In this shorter form we can solve both inequalities
at once by performing the same operation on all three
parts of the inequality. The goal is to isolate the x in
the middle.
Example
Solve and graph the solution set on a number line.
3  x  1  2
y




x










Solving Inequalities
with Absolute Value
The graph of the solution set for x >c will be divided
into two intervals whose union cannot be represented as
a single interval. The graph of the solution set for x  c
will be a single interval. Avoid the common error of rewriting
x  c as -c<x>c.
Example
Solve and graph the solution set on a
number line.
2x  5  3
Example
Solve and graph the solution set on a
number line.
2x  5  3
Applications
Example
A national car rental company charges a flat rate of
$320 per week for the rental of a 4 passenger sedan.
The same car can be rented from a local car rental
company which charges $180 plus $ .20 per mile.
How many miles must be driven in a week to make
the rental cost for the national company a better
deal than the local company?
Solve the absolute value inequality.
2 x2  4
4  x  0
(b) x  4 or x  0
(c) x  4 and x  0
(d) x  4 or x  0
(a)
Solve the linear inequality.
4  x  3  6x  9
3
(a) x  
2
3
(b)
x
2
(c) x  3
(d) x  2