1. Absolute Value Inequalities

2. G. O. L. A. < Less Than > Greater Than is an is an “and” “or”
Statement
Graphs must overlap
Thumbs in
<---|---|---|---|---|---|---|---|---|--->
> Greater Than
is an
“or”
Statement
Graphs go different directions.
Thumbs out
<---|---|---|---|---|---|---|---|---|--->

3. Remember, ‘and’ statements can be written as a compound sentence X > -1 and X < 5
Turn this one
around
Bring this one
down
Drop the word “and”
NOW
PUT
THEM
TOGETHER

4. Don’t forget to flip the symbol And one for the POSITIVE
Now what
2 numbers
will give us
this 2?
L. A. Less = And
1) | x + 4 | ≤ 2
Now find the value of “x”, set up 2 inequalities using the information found between the absolute value bars…..
BUT FIRST – what happens to the inequality symbol when you multiply or divide by a negative number?
You flip it
One for the NEGATIVE
Don’t forget to flip the symbol
And one for the POSITIVE
DON’T FLIP!!!
so...
Now
put them
together

5. Let’s graph -6 ≤ x ≤-2 <---|---|---|---|---|---|---|---|---|--->
What kind of
Circles?
Now shade
Between the
circles !!
Let’s graph -6 ≤ x ≤-2
<---|---|---|---|---|---|---|---|---|--->
When graphing COMPOUND INEQUALITIES
( the AND statements ), shading stops at the endpoints.
Another way to say it:
X is between -6 and -2
So….
Shade between -6 and -2

6. 2) | x - 9 | ≤ 5 so... less than is an “and” statement
What kind of statement?
2) | x - 9 | ≤ 5
Now what 2 numbers
will give us this 5?
less than is an “and” statement
thumbs in to graph
Set up the NEGATIVE side
Don’t forget to flip the symbol
And set up the POSITIVE side
DON’T FLIP!!!
so...
Now
put them
together

7. What kind of
Circles?
Now shade
Between the
circles !!
Let’s graph it.. 4 ≤ x ≤14
<---|---|---|---|---|---|---|---|---|--->
When graphing COMPOUND INEQUALITIES
( the AND statements ), shading stops at the endpoints.
Another way to say it:
X is between 4 and 14
So….
Shade between 4 and 14

8. What kind of statement?
3) | 4x – 5 | < 3
You
try it!
less than is an “and” statement – thumbs in
Set up the NEGATIVE side
Don’t forget to flip the symbol
And set up the POSITIVE side
DON’T FLIP!!!
so...
Now
put them
together

9. Now shade
Between the
circles !!
What kind of
Circles?
Graph it ½ < x < 2
<---|---|---|---|---|---|---|---|---|--->
When graphing COMPOUND INEQUALITIES
( the AND statements ), shading stops at the endpoints.
Another way to say it:
X is between ½ and 2
So….
Shade between ½ and 2

10. What about this one? so... 4) | x | > 4
What kind of statement?
4) | x | > 4
greater than is an “or” statement
Remember, “or” in opposite directions – Thumbs out
Set up the NEGATIVE side
Don’t forget to flip the symbol
And set up the POSITIVE side
DON’T FLIP!!!
Or’s
are not compound
statements!
Don’t put them
together.
so...
Graph
<---|---|---|---|---|---|---|---|---|--->

11. You try it
5) | 2x + 1 | ≥ 8
What kind of statement?
greater than is an “or” statement - Thumbs out
Set up the NEGATIVE side
Don’t forget to flip the symbol
And set up the POSITIVE side
DON’T FLIP!!!
so...

12. Now graph it! "or" - thumbs out -5 -4 -3 -2 -1 0 1 2 3 4 5 6
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
"or" - thumbs out

13. Your turn so... Now graph it.. -16 -14 -12 -10 -8 -6 -4 -2 0 2
What kind of statement?
6) | x + 9 | ≥ 7
Set up the NEGATIVE side
Don’t forget to flip the symbol
And set up the POSITIVE side
DON’T FLIP!!!
so...
Now graph it..
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->

14. A POSITIVE IS ALWAYS GREATER THAN A NEGATIVE Therefore
7) | x + 9 | ≥ -5
The absolute value of any number is positive, therefore
| x + 9 | ≥ -5
+ ≥ -
A POSITIVE IS ALWAYS GREATER THAN A NEGATIVE Therefore
ALL REAL NUMBERS
Is the answer!!!
Absolute
Value Is
Pos.
True

15. A POSITIVE IS NEVER LESS THAN A NEGATIVE Therefore
8) | x - 6 | ≤ -2
The absolute value of any number is positive, therefore
| x - 6 | ≤ -2
+ ≤ -
A POSITIVE IS NEVER LESS THAN A NEGATIVE Therefore
NO SOLUTION
Is the answer!!!
Absolute
Value Is
Pos.
False