Luttrell 2012
77
Name: ______________________
Date: _____
5d – Absolute Values
On a real number line, what is the distance between: 5 and 12, -3 and 5, or 4 and 7? We find the
distance by subtracting the smaller value from the larger: 12-5, 5 - (-3), 7 - 4. So what is the
distance between 0 and
x
? It would be written as
x
- 0. But what if we knew
x
was 5 units from
zero? The number could be five less or five more than zero. Then symbolically it would be
written as |
x
- 0| = 5, then
x
is either -5 or 5. The absolute value,
magnitude
, of
x
- 0 (or
x
) gives
the distance, without specifying direction.
The equation |
x
- 1| = 5, can be thought as the distance
x
from 1 is 5 units. So starting on the real
number line at 1, you would count to the right or left 5 units, getting two answers: 6 and - 4.
E
x
pressions with absolute values can get more complicated, so you may want to remember a
certain rule: |
x
y| = |
x
|
⋅
|y|. For e
x
ample: |-
x
| = 3 can be written as |-1|
⋅
|
x
| = 3. Then |-1| really is 1
since the absolute value is asking for the magnitude of -1. So the equation really is 1
⋅
|
x
| = 3,
which gives the answer of -3 and 3.
Simplify each expression and graph the solution.
1. The distance between 4 and 6 is 2.
2. The distance between 5 and 11 is 6.
3. The distance between 7 and -3 is 10.
4. The distance between x and 3 is 5.
5. The distance between x and -2 is 4.
6. The distance between x and 3 is more than 6.
7. The distance between x and -2 is less than 4.
Solve each equation by translating its symbolic meaning first. Graph the solution.
8. | x - 4 | = 2
9. | x - 8| = 3
10. | x + 3| = 5
11. |x + 2| = 6
