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```3.6 Abs Value Equations
Hornick
Objectives
3.6 Solve equations in one variable that contain
absolute-value expressions.
Remember...
Recall that the absolute-value of a number is
that number’s distance from zero on a number
line. For example, |–5| = 5 and |5| = 5.
5 units
6 5 4 3 2 1
5 units
0
1
2
3
4
5
6
Isolate absolute value first!
Then you must consider two cases…
Example 1
Solve the equation.
|x| = 12
|x| = 12
Think: What numbers are 12 units
from 0?
•
12 units
12 10 8 6 4 2
Case 1
x = 12
Case 2
x = –12
•
0
12 units
2
4
6
•
8 10 12
Rewrite the equation as two
cases.
The solutions are {12, –12}.
Example 2
Solve the equation.
3|x + 7| = 24
|x + 7| = 8
Case 1
Case 2
x + 7 = 8 x + 7 = –8
– 7 –7
–7 –7
x
=1 x
= –15
The solutions are {1, –15}.
Not all absolute-value equations have two solutions.
• If the absolute-value expression equals zero, there
is one solution.
• If an equation states that an absolute-value is
negative, there are no solutions.
Example 3
Solve the equation.
8 = |x + 2|  8
8 = |x + 2|  8
+8
+8
0 = |x + 2|
0= x+2
2
2
2 = x
The solution is {2}.
There is only one case!
Example 4
Solve the equation.
3 + |x + 4| = 0
3 + |x + 4| = 0
3
3
|x + 4| = 3
Absolute value cannot be
negative! You can’t have a
negative distance!
This equation has no solution.
You can write
 to show that this set is empty.
Remember!
Absolute value must be nonnegative because it
represents a distance.
Grab a whiteboard, marker, and eraser per pair!
Solve the following equation. Show all work!!
3 = |x + 4|  8
Solve the following equation. Show all work!!
16 = 5|x |  4
Solve the following equation. Show all work!!
 2|7x | = 14
Solve the following equation. Show all work!!
2|x+3 | = 8
Solve the following equation. Show all work!!
1.2|5x| = 3.6
Solve the following equation. Show all work!!
 2|x| = 3
Solve the following equation. Show all work!!
 2|5x| =  3
Challenge
Solve the following equation. Show all work!!
|x+4| = 3x
Classwork/Homework
3.6 Practice Worksheet
Exit Card
Solve each equation.
1. 15 = |x|
3. |x + 1|– 9 = –9
5. 7 + |x – 8| = 6
2. 2|x – 7| = 14
4. |5 + x| – 3 = –2
Exit Card
Solve each equation.
1. 15 = |x| {–15, 15} 2. 2|x – 7| = 14 {0, 14}
3. |x + 1|– 9 = –9 {–1}4. |5 + x| – 3 = –2 {–6, –4}
5. 7 + |x – 8| = 6
no solution
```
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