Download How to write a solution set If the shading is on the outside of the number  line like: 

(3) Absolute Value Inequalities.notebook
How to write a solution set
If the shading is on the outside of the number line like: ­8
­5
Your solution set will use the word "or"
If the shading is in between the numbers on the number line like: ­8
­5
Your solution set will be written with no
words and the inequality signs will face the
same direction!
­8<x<5
(this inequality means "and")
Absolute Value Inequalities
|x + 2| < 2
To solve you must:
1. Isolate the absolute value.
2.
Treat the inequality sign as an equal sign and solve as you would an absolute value equation. 3.
Once you have your two answers, place them in numerical order on a number line. < > open circles
≤≥ closed circles 4.
Write your answers as a solution set.
“great ­ or than” (Solution has the word or)
“less th­and” (Solution is written with x in the middle)
INEQUALITY SHORT CUT: Once the absolute value is isolated, if the inequality is : (OR) “greater than” ­ shade the outside of the number line.
(AND) “less than” ­ shade in between the numbers
(3) Absolute Value Inequalities.notebook
Example 1: (solving with "greater than") OR
Solve:
Case 2:
Case 1:
=
=
=
=
Note that there are two parts to the solution and that the connecting word is "or". 25
15
{x < 15 or x > 25}
Example 2: (solving with "less than or equal to")
Solve:
x ­ 3 = 4 x = 7
x ­ 3 = ­4
x = ­1
­1
7
{­1 ≤ x ≤ 7}
AND
(3) Absolute Value Inequalities.notebook
Example 3: (isolating the absolute value first)
Solve:
|3 + x| < 4
3 + x = ­4
x = ­7
3 + x = 4
x = 1
­7
1
{­7 < x < 1}
Example 4: Solve: 3 d + 1 + 6 ≥ 24
3|d + 1| ≥ 18
|d + 1| ≥ 6
d + 1 = 6
d = 5
d + 1 = ­6
d = ­7
­7
5
{d ≤ ­7 or d ≥ 5}
Example 5: (all values work)
Solve:
abs. value must be positive,
so it will always be > ­3
x is all real numbers (the whole number line is shaded)
Example 6: (no values work)
Solve:
abs. value must be positive,
so it will never be < ­6
no solution (nothing is shaded on the number line)
(3) Absolute Value Inequalities.notebook
Example 7: Solve:
|5 ­ 4x | < 4
5 ­ 4x = ­4
­4x = ­9
x = 9
5 ­ 4x = 4
­4x = ­1
x = 1
4
4
1
4
{
9
4
}