Transcript Absolute Value - hancockhighmath

Absolute Value Inequalities

G. O.
L. A.
> Greater Than
is an
“or”
Statement
< Less Than
is an
“and”
Statement
Graphs go different
directions.
 Thumbs out
<---|---|---|---|---|---|---|---|---|--->

Graphs must overlap
 Thumbs in
<---|---|---|---|---|---|---|---|---|--->
Remember, ‘and’ statements can be
written as a compound sentence
X > -1 and X < 5
Bring this one
down
Turn this one
around
Drop
the
word
“and”
1  x x  5
NOW
PUT
THEM
TOGETHER
1  x  5
1) | x + 4 | ≤ 2
Now what
2 numbers
will give us
this 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?
One for the NEGATIVE
And one for the POSITIVE
Don’t forget to flip the symbol
DON’T FLIP!!!
x  4  2
4 4
x  6
Now
put them
together
x4 2
4 4
x  2
x  6 and x  2
6  x  2
Let’s graph
-6 ≤ x ≤-2
<---|---|---|---|---|---|---|---|---|--->
-8 -7
-6 -5 -4
-3 -2
-1
0
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
What kind of
statement?
2) | x - 9 | ≤ 5
less than is an “and” statement
thumbs in to graph
Now what 2 numbers
will give us this 5?
Set up the NEGATIVE side
And set up the POSITIVE side
Don’t forget to flip the symbol
DON’T FLIP!!!
x  9  5
9 9
x 4
Now
put them
together
x 9  5
9 9
x  14
x  4 and x  14
4  x  14
Let’s graph it..
4 ≤ x ≤14
<---|---|---|---|---|---|---|---|---|--->
2
4
6
8 10 12 14 16 18
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
3) | 4x – 5 | < 3
What kind of
statement?
less than is an “and” statement – thumbs in
Set up the NEGATIVE side
And set up the POSITIVE side
Don’t forget to flip the symbol
DON’T FLIP!!!
4 x  5  3
5 5
4x  2
4
4
4x  5  3
5 5
4x  8
4
4
2
1
x
or x 
4
2
Now
1
put them
x  and x  2
together
2
x2
1
x2
2
Graph it
½<x<2
<---|---|---|---|---|---|---|---|---|--->
0 1 2 3
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
4)
What kind of
statement?
|x|>4
greater than is an “or” statement
Remember, “or” in opposite directions – Thumbs out
Set up the NEGATIVE side
And set up the POSITIVE side
Don’t forget to flip the symbol
DON’T FLIP!!!
x4
x  4
x  4 or x  4
Graph
<---|---|---|---|---|---|---|---|---|--->
-4 -2
0
2
4
Or’s
are not compound
statements!
Don’t put them
together.
5) | 2x + 1 | ≥ 8
greater than is an “or” statement - Thumbs
What kind
of
statement?
out
Set up the NEGATIVE side
And set up the POSITIVE side
Don’t forget to flip the symbol
DON’T FLIP!!!
2x 1  8
1 1
2x  9
2 1
9 2
x
2
or x  4
2
2x  1  8
1 1
2x  7
2
2
1
7
x  or x  3
2
2 1
1
x  4 or x  3
2
2
1
1
x  4 or x  3
2
2

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
What kind of
statement?
6) | x + 9 | ≥ 7
Set up the NEGATIVE side
And set up the POSITIVE side
Don’t forget to flip the symbol
DON’T FLIP!!!
x9  7
9 9
x  2
x  9  7
9 9
x  16
x  16 or x  2

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-16
-14 -12 -10 -8
-6
-4 -2
0
2
7) | x + 9 | ≥ -5

Absolut
e
Value
Is
Pos.
The absolute value of any number is
positive, therefore
| x + 9 | ≥ -5
+≥ -
8) | x - 6 | ≤ -2

Absolut
e
Value
Is
Pos.
The absolute value of any number is
positive, therefore
| x - 6 | ≤ -2
+≤ -