Directed Distance

1
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
The inside could have been or 13 so need to
solve both!
2
Distance Between Two Points.

• What is the distance between the two values of 10
  and 2?
• What is the distance between the two values of
  -102 and 80?
• So the distance between two points x1 and x2 is
  x1 x2 or x2 x1

8
182
3
Directed Distance

• The directed distance from a to b is b a.
• Ex Find the directed distance from 5 to -10
• -10 5
• -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 down, so -15
-10
5
Had to go up, so 15
4
Midpoint

• The midpoint between to values is a b
• 2
• Ex Find the midpoint of the interval 1, 10
• 110
• 2
• 5.5

10
5
6
0
1
5
Absolute Value
Is this statement true?
Not true
6
Absolute Value
Think of absolute value as measuring a distance.
7
Absolute Value
Absolute Value The distance a number is from
zero on a number line.
It is always positive or zero.
8
Absolute Value
(
)
The lt sign indicates that the value is center
around 0 and no more than 3 away.
9
Absolute Value
Now the subtraction of 2 has translated our
center to 2.
(
)
The lt sign indicates that the value is centered
around 2 and no more than 3 away.
NOTICE 2 is the midpoint of -1 and 5.
10
Absolute Value


The gt sign indicates that the value is diverging
from points on either side of 0.
11
Absolute Value
Now the subtraction of -3 has translated our
center to -3.


The gt sign indicates that the value is diverging
from points on either side of -3.
NOTICE -3 is the midpoint of -4 and -1.
12
Writing an Absolute Value
Ans x4

1. Write an absolute value inequality for the below
   intervals(-8, - 4U4, 8) (-5, 5) (-
   8, 2)U(5, 8) -10, 20

Ans xlt5
Ans x 3.5gt1.5
Ans x 515
13
Absolute Value
What does this statement mean?
(
)
14
Absolute Value
What does this statement mean?


15
Absolute Value
Think of what we just saw. This picture would
have two pieces, since the distance is greater.
16
Absolute Value
Think of what we just saw. This picture would
have one piece between two numbers, since the
distance is smaller.
17
You Try

• Solve the following inequalities
• 2xlt 6
• 3x14
• 25 xgt20

Ans (-3, 3)
Ans (-8,-5/3 U 1,8)
Ans (-8,5) U (45,8)
18
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.