Solving Linear Inequalities
Chapter 2, Sections 5
 Properties Used to Solve Inequalities
 For real numbers a, b, and c:
 If a > b, then a + c > b + c.
 If a > b, then a  c > b  c.
 If a > b and c > 0, then ac > bc.

If a > b and c > 0, then 
a c 
> 
b c 
 If a > b and c < 0, then ac < bc.

If a > b and c < 0, then 
a c 
< 
b c 
 Note that when you multiply by a negative number the inequality sign changes direction.
 How to Write Solutions

In Set Builder Notation

On Number Line

In Interval Notation

x > a 
{ x  x > a } 

(a, ) 
x a 
{ x  x a } 

[a, ) 
x < a 
{ x  x < a } 

(, a) 
x a 
{ x  x a } 

(, a] 
a < x < b 
{ x  a < x < b } 

(a, b) 
a x b 
{ x  a x b } 

[a, b] 
a < x b 
{ x  a < x b } 

(a, b] 
a x < b 
{ x  a x < b } 

[a, b) 
 Solving Simple Inequalities
 Solve a simple linear inequality like you would solve a linear equation except that the = sign is repaced by an inequality sign.

Example 
Solution Steps 
Solve 5x  2 < 3 
5x < 5
x < 1 
The solution is x < 1

 Solving Compound Inequalities
 A compound inequality has an expression in the middle sandwiched between two inequalities
like in 5 < 2x  3 11.
 Goal: get x alone (with coefficeint of 1) in the middle of the sandwich with numbers on the outsides.
 What you do to the middle (while trying to get x alone) you must do to the outside expressens as well. That will keep each step's inequality equivalent to the original inequality (equivalent means that the solutions are the same).

Example: Solve 5 < 2x  3 < 11

Solution Steps 
In Words... 
8 < 2x < 14 
Add 3 to each expression. 
4 < x < 7 
Divide each expression by 2. 
The solution is 4 < x < 7


