Download Step 4

x3  9
Solve Absolute
Value Inequalities
2x4 8
© 2011 The Enlightened Elephant
First, let’s think about
what absolute value
inequalities are really
asking us to find!
© 2011 The Enlightened Elephant
What does
x 4
x 4
All the numbers whose distance
from zero is greater than 4.
4
-4
x  4
really mean?
or
x4
***Notice that you need to have two inequalities
to represent the distances that are greater than
4 from zero.
© 2011 The Enlightened Elephant
What does
x 4
x 4
really mean?
All the numbers whose distance
from zero is less than 4.
x  4
-4
and
4
x4
***Notice that you need to have two inequalities
to represent the distances that are less than 4
from zero.
However, since these inequalities must happen at
the same time, it should be written as
© 2011 The Enlightened Elephant
4 x  4
OK, so how do we solve
more difficult problems?
© 2011 The Enlightened Elephant
Solve and graph 2 x  4  8
Step 1: Isolate the absolute value.
x4  4
Step 2: Set up two inequalities.
x4  4
x  4  4
Step 3: Solve the inequalities.
x 8
x0
Step 4: Graph the solutions.
0
© 2011 The Enlightened Elephant
8
Let’s practice!
© 2011 The Enlightened Elephant
You Try!
Step 1: Solve and graph. 4 x  2
Step 2:
Step 3:
Step 4:
 1  11
x2 3
x23
x  2  3
x 1
x  5
Step 5:
-5
Final answer: -5<x<1
© 2011 The Enlightened Elephant
1
You Try!
Set up two
inequalities!
x3  9
x  3  9
x 3  9
Solve and graph.
x6
x  12
-12
Final answer:
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6
x6
or
x  12
Isolate the absolute
value first!
You Try!
Solve and graph.
3 2 x  1  21
2x 1  7
2 x 1  7
x4
2x 1  7
x  3
-3
Final answer:
© 2011 The Enlightened Elephant
4
x4
or
x  3
Isolate the absolute
value first!
You Try!
Solve and graph.
3x  6  3  15
3x  6  12
3x  6  12 3x  6  12
x6
-2
Final answer:
© 2011 The Enlightened Elephant
x  2
6
2 x  6
Isolate the absolute
value first!
You Try!
Solve and graph.
3 2 x  1  8  23
2x 1  5
2x  1  5
x2
-3
Final answer:
© 2011 The Enlightened Elephant
2x 1  5
x  3
2
3  x  2
You Try!
Solve and graph.
Isolate the absolute
value first!
2 x  8  12
Final answer: NO SOLUTION!
Absolute values are distances, which
cannot be negative.
You can choose any value for x and the left
side of the inequality will always be
positive.
However, positive numbers are NEVER
less than negative numbers.
© 2011 The Enlightened Elephant
You Try!
Solve and graph.
Isolate the absolute
value first!
6x 12  10
Final answer: ALL REAL NUMBERS!
Absolute values are distances, which
cannot be negative.
You can choose any value for x and the left
side of the inequality will always be
positive.
Positive numbers are ALWAYS greater
than negative numbers.
© 2011 The Enlightened Elephant
Success!
Nice job!
© 2011 The Enlightened Elephant