Math 101 Intermediate Algebra

Solving Linear Inequalities
Chapter 2, Sections 5

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 ac > bc

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

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

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