Download absolute value equations

Absolute Value Equations
Objectives:
…to graph an absolute value equation (function) on the coordinate plane
...to solve absolute value equations
Assessment Anchor:
Not Available at this time.
NOTES and EXAMPLES
We’ve focused our attention this year on working with linear equations. Now, we
want to give you your first taste of something that is NOT linear.
As we investigate what happens for an equation with an absolute value part, we
find that the graph for these kinds of equations is actually in the shape of a “V.”
That “V” can be right side up, or upside down. That “V” can be awfully skinny, or
awfully fat, or anywhere in between.
The highest or lowest point of that “V” is called the vertex. And, it is the vertex
that we are interested in finding first.
***To graph an absolute value equation:
1. Find the x-coordinate of the vertex
by setting what’s inside the absolute
value part equal to zero
2. Make a T-table of 5 ordered pairs
(the x-coordinate of the vertex, and
two values above and below it)
3. Plot all 5 points and sketch the
V-shaped graph
1)
y=|x–4|
x–4=0
+4+4
x=4
vertex-->
x
2
3
4
5
6
y
2
1
0
1
2
Absolute Value Equations
2)
y = | 3x | – 4
x
y
x
y
vertex
3)
y = -4| x | + 5
vertex
4)
y = 1 + | 2x – 4 |
x
vertex
y
Absolute Value Equations
***To solve an absolute value equation:
1.
Isolate the absolute value part!
2. Determine number of solutions
a. If the absolute value part is equal to a negative #...
NO SOLUTION!!
b. If the absolute value part is equal to a positive #...
TWO SOLUTIONS!!
3. If there are solutions, make linear equations (no absolute value
symbols) and solve each one
5a)
| x – 4 | + 5 = 12
5b)
2| x + 9 | = 22
6b)
4| 2x + 1| = 12
| x – 4 | + 5 = 12
Isolate --->
–5 –5
Examine --->
|x–4|=7
Two solutions!
x–4=7
and
x – 4 = -7
+4 +4
+4 +4
x = 11
x = -3
x = 11 and x = -3
6a)
| x – 6 | + 11 = 9
| x – 6 | + 11 = 9
Isolate --->
– 11 – 11
Examine --->
| x – 6 | = -2
NO SOLUTION!
Absolute Value Equations
7)
-3| 2x – 1 | + 8 = 2
8)
| -3x – 5 | = 16
9)
| 2x + 4 | – 13 = -1
10)
-2| 12 x + 7 | – 9 = -1