section 1.7 linear inequalities and absolute value inequalities
TRANSCRIPT
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 b
oth sides of an inequality by
a negative number, the direction of the inequality symbol
is reversed.
Checking the solution of a linear inequality on a Graphing Calculator
1
2
2 1
4
y x
y x
2 1 4x x
Y1=2x+1
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.
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.
Separate the inequality into two equations.
The solution set could be the null set, . The solution
set could be all real numbers, - , .
1
0 1
Never true
x x
1
0 1
Always true ,
x x
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 t
hree
parts of the inequality. The goal is to isolate the x in
the middle.
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
will be a single interval. Avoid
c
the common error of rewriting
x as -c<x>c.c
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?