Download Absolute Value

Directed Distance &
Absolute Value
Objective: To be able to find directed distances and
solve absolute value inequalities.
TS: Making Decisions after Reflection and Review
Warm Up: Solve the following absolute value
equation.
|2x – 3| = 13
2 x  3  13 or 2 x  3  13
2 x  16 or 2 x  10
x  8 or x  5
The inside could
have been+ or –
13 so need to
solve both!
Distance Between Two Points.
What is the distance between the two
values of 10 and 2? 8
What is the distance between the two
values of -102 and 80? 182
So the distance between two points x1 and
x2 is |x1 – x2| or |x2 – x1|
Directed Distance
The directed distance from a to b is b – a.
Ex: Find the directed distance from 5 to -10
-10 – 5
-10
-15
5
Had to go down, so -15
The directed distance from b to a is a – b.
Ex: Find the directed distance from -10 to 5
5 – (-10)
15
-10
5
Had to go up, so +15
Midpoint
The midpoint between to values is a + b
2
Ex: Find the midpoint of the interval [1, 10]
1+10
2
5.5
0
1
5
6
10
Absolute Value
Is this statement true?
A B  A  B
Not true
5  3
5  3
2
53
2
8
Absolute Value
Think of absolute value as measuring a distance.
0
5
Absolute Value
Absolute Value: The distance a number is from zero on a number line.
3
0
It is always positive or zero.
5
Absolute Value
A 3
(
3
)
0
3
The < sign indicates that the value is center around 0 and no more than 3 away.
Absolute Value
A2  3
Now the subtraction of 2 has “translated” our center to 2.
(
1
)
2
5
The < sign indicates that the value is centered around 2 and no more than 3 away.
NOTICE: 2 is the midpoint of -1 and 5.
Absolute Value
A 2
]
2
[
0
2
The > sign indicates that the value is diverging from points on either side of 0.
Absolute Value
A3  2
Now the subtraction of -3 has “translated” our center to -3.
]
4
[
3
1
The > sign indicates that the value is diverging from points on either side of -3.
NOTICE: -3 is the midpoint of -4 and -1.
Writing an Absolute Value
1)
Write an absolute value inequality for the
below intervals:
(-∞, - 4]U[4, ∞)
Ans: |x|≥4
(-5, 5)
Ans: |x|<5
(- ∞, 2)U(5, ∞)
Ans: |x – 3.5|>1.5
[-10, 20]
Ans: |x – 5|≤15
Absolute Value
What does this statement mean?
A B
(
B
)
0
B  A  B
B
Absolute Value
What does this statement mean?
A B
]
B
[
0
A   B or A  B
B
Absolute Value
A  B can be converted to  B  A  B
A  B can be converted to A   B or A  B
3x  1  7
3x  1  7
or
3x  1  7
3 x  8
3x  6
x   83
x2
(,  83 ]  [2, )
Think of what
we just saw.
This picture
would have
two pieces,
since the
“distance” is
greater.
Absolute Value
A  B can be converted to  B  A  B
A  B can be converted to A   B or A  B
6x  4  1
1  6 x  4  1
4
4 4
5  6 x  3
6
6
6
 56  x   12
  56 ,
 12 
Think of what
we just saw.
This picture
would have
one piece
between two
numbers, since
the “distance”
is smaller.
You Try
Solve the following inequalities:
1) |2x|< 6
Ans: (-3, 3)
2)
|3x+1|≥4
Ans: (-∞,-5/3] U [1,∞)
3)
|25 – x|>20
Ans: (-∞,5) U (45,∞)
Conclusion
• Absolute value is the distance a number is
from zero on a number line.
• Two equations are necessary to solve an
absolute value equation.
• Two inequalities are necessary to solve an
absolute value inequality.