Download Solving Linear Inequalities

Avon High School
Section: 1.7
ACE COLLEGE ALGEBRA II - NOTES
Linear Inequalities and Absolute Value Inequalities
Mr. Record: Room ALC-129
Day 1 of 1
Interval Notation and Set-Builder Notation
Let a and b be real numbers such that a < b.
Interval
Notation
Set-Builder Notation
Graph
 a, b 
a, b
 a, b 
 a, b 
 a,  
a, 
 ,b 
 ,b
 ,  
Solving Linear Inequalities in One Variable
Example 1
a.
1,3  2,6
Finding Intersections and Unions of Intervals
Use graphs to find each set.
b.
1,3  2,6
Investigation: Consider the following true inequality statement: 2 < 5.
Perform the following operations to both sides and determine if the
statement remains true or becomes false.
(i) Add 4 to both sides
T
F
(ii) Subtract 1 from both sides
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F
(iii) Multiple both sides by 2
T
F
(iv) Multiply both sides by -3
T
F
(v) Divide both sides by -2
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Example 2
Solving a Linear Inequality
a. Solve 2  3x  5
b. Solve 3x 1  7 x 15
Investigation: For Example 2 – part b, use a graphing calculator to
sketch each side of the inequality as an equation with y by itself.
How could you determine the solution to the inequality by looking at the
resulting graphs of the two equations?
Example 3
Solving a Linear Inequality Containing Fractions
Solve and graph the solution set on a number line:
x4 x2 5

 .
2
3
6
Inequalities With Unusual Solution Sets
Example 4
a.
Solving Linear Inequalities
Solve each inequality.
3( x  1)  3 x  2
b.
x  1  x 1 .
Solving Compound Inequalities
Example 5
Solving a Compound Inequality
Solve and graph the solution set of the following inequality on a number line.
1  2x  3  11
This inequality could also be rewritten as:
Solving Inequalities with Absolute Value
Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number,
1. The solutions of u  c are the real numbers that satisfy c  u  c.
2. The solutions of u  c are the real numbers that satisfy u  c or u  c.
To better understand
these rules, stop and
think about what the
solution to |x| < 2 is.
These rules are valid if < is replaced by ≤ and > is replaced by ≥.
Example 6
Solving an Absolute Value Inequality
Solve and graph the solution set of each inequality on a number line.
a.
x2 5
c. 18  6  3x
b. 3 5x  2  20  19
Applications
Example 7
Selecting the Better Deal
Rent-A-Heap car rental agency charges $4 a day plus $0.15 per mile, whereas Clunkers-R-Us rental
agency charges $20 a day and $0.05 per mile. How many miles must be driven to make the daily cost of
a Rent-A-Heap rental a better deal that a Clunkers-R-Us rental?