Intermediate Algebra chapter 2 part 4

Solving Linear Inequalities
Chapter 2,

Properties Used to Solve Inequalities

For real numbers a, b, and c:

1. If a > b, then a + c > b + c.
2. If a > b, then a - c > b - c.
3. If a > b and c > 0, then ac > bc.

4. If a > b and c > 0, then

   a c

   >

   b c

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

6. 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