This document covers solving and graphing linear inequalities and absolute value inequalities in one variable. It discusses: - Solving linear inequalities by reversing the inequality sign if multiplying or dividing by a negative number. - Graphing the solutions of inequalities on a number line, using open or closed circles to indicate greater than, less than, greater than or equal to, and less than or equal to. - Defining absolute value as the distance from a number to zero on the number line. - Converting absolute value inequalities to equivalent linear inequalities that can be solved algebraically. - Worked examples of solving and graphing different types of inequalities.

1. Ch. 3 – Inequalities
3.1 Linear Inequalities;
Absolute Value
Objectives:
Solve and graph linear inequalities in one variable

2. Solving Inequalities
• Solve just like an equation, EXCEPT:
• The inequality sign must be reversed if you
multiply or divide by a negative number
Graphing solutions
• On a number line:
• < , > use open circle
•  ,  use closed circle
• Shade (or use an arrow) to indicate solution
set.

3. Example 1a
• Solve 3x – 4  10 + x and graph the
solution.

4. Example 1b
• Solve and graph the solution.

5. You Try!
• Solve and graph the
solution.

6. Absolute Value
• |x| means (geometrically) the distance
from x to zero on the number line. (c  0)
Sentence Meaning
The distance
from x to 0 is:
Graph Solution
|x| = c exactly c units x = c or x = -c
-c 0 c
|x| < c less than c units -c < x< c
|x| > c greater than c
units
x < -c or x > c
-c 0 c
-c 0 c

7. • Sentences with |x – k| can mean the
distance from x to k on the number line.
Sentence Meaning Graph Solution
|x - 5| = 3 The distance
from x to 5 is 3
units
x = 2 or x = 8
|x - 1| < 2 The distance
from x to 1 is
less than 2 units
-1 < x< 3
|x + 3| > 2 or
|x – (-3)| > 2
The distance
from x to -3 is
greater than 2
units
x < -5 or x > -1

8. Algebraic Method
Sentence Equivalent Sentence
|ax + b| = c ax + b = c
|ax + b| < c -c < ax + b < c
|ax + b| > c ax + b < -c or ax + b > c

9. Example 2a
• Solve |3x – 9| > 4 and graph the solution.

10. Example 2b
• Solve |2x + 5|  7 and graph the solution.

11. You Try!
• Solve and graph the solution:
• |2x + 3 | = 1
• |x – 2|  3