Download Solving Absolute Value Sentences Graphically

Name __________________________________
Period __________
Date:
Topic: 2-5 Solving
Absolute Value
Sentences
Graphically
Standard: F-IF.7b
Objective:
Essential Question: When would you use a graphical
technique rather than an algebraic technique to solve an
absolute value open sentence?
Graph square root, cube root, and piecewise defined functions, including
step functions and absolute value functions.
To use number lines to obtain quick solutions to certain
equations and inequalities involving absolute value.
You know that on a number line the distance between the
graph of a real number x and the origin is | |. The distance on
|, or
the number line between real numbers a and b is |
|
|.
equivalently,
You can use this fact to solve many
open sentences almost at sight.
|𝒂
𝒃|
a
b
|𝒃
Summary
𝒂|
Find the distance between the graphs of 8 and 17.
Example 1:
Solution:
|
|
|
|
Find the distance between the graphs of 11 and 6.
Solution:
|
(
)|
|
|
|
|
Find the distance between the graphs of 9 and 12.
Solution:
Exercise 1:
|
|
|
|
Find the distance between the graphs of 9 and 16.
Find the distance between the graphs of 11 and 16.
Find the distance between the graphs of 9 and 25.
2
Example 2:
Solve graphically
|
|
|
Solution: To satisfy |
, x must be a number whose distance
from 1 is 2 units. So, to find x, start at 1 and move 2 units in
each direction on a number line.
Start
2
2
4
Exercise 2:
3
2
1
0
1
2
3
4
-4- -3- -2- -1- 0
1
2
3
4
Solve graphically
|
|
Solve graphically
|
|
-4 -3 -2 -1 0
1
2
3
4
3
Solve graphically
Example 3:
|
|
(
Solution:
)
|
( )|
Therefore, |
, is equivalent to |
. So
the distance between y and 1 must be 3 units or less. To find
y, start at 1 and move 3 units in each direction on a number
line.
Start
3
-4 -3 -2 -1 0
3
1
2
3
4
The numbers up to and including 2 and the numbers down to
and including 4 will satisfy the inequality.

Exercise 3:
the solution set is *
+
Answer
Solve graphically
|
|
-4 -3 -2 -1 0
1
2
3
4
4
Certain equations and inequalities, such the ones in the
previous examples and exercises, lend themselves easily to a
graphic solution. With these types, the expression involving
|. When the
absolute value is of the form |
expression is more complicated, as in the following example, a
graphic method may still be applied, though not as easily. For
these types, you might prefer to use the algebraic method you
learned in the previous lesson.
Example 4:
Solve graphically
|
|
Solution: Use the facts that |
|
rewrite |
|
|
|
| |
as follows:
|
|
| and |
(
|
| | | | to
)|
| | |
|
|
|
Therefore, the given inequality is equivalent to,
|
|
|
or
|
To find t, start at and move more than to the right and to the
left.
Start
-4 -3 -2 -1 0
1
2
3
4
5
Exercise 4:
Solve graphically
|
|
-4 -3 -2 -1 0
1
2
3
4
Now, solve the same inequality algebraically.
Class work:
p 78 Oral Exercises: 1-24
Homework:
p 78 Written Exercises: 2-24 even
p 79 Mixed Review: 1-9
6