Download Solving Open Sentences Involving Absolute Value

7-6: Solving Open Sentences Involving Absolute Value
OBJECTIVE:
You will be able to solve open sentences involving absolute value and
graph the solutions.
We need to start with a discussion of what absolute value means.
Absolute value is a means of determining the distance from zero.
If I ask for the absolute value of 4, the answer is “4” since 4 is four units
away from zero.
If I ask for the absolute value of -4, the answer is “4” since -4 is four
units away from zero.
|4| = 4
|-4| = 4
© William James Calhoun, 2001
7-6: Solving Open Sentences Involving Absolute Value
Now examine absolute values with variables in them.
|x| = 4 can be rewritten as:
|x - 0| = 4
Rewriting the absolute value this way helps to explain what is going on.
The absolute value tells us, “The distance between x and zero is four
units.”
So, x if four units away from zero in either direction.
Graphically:
In non-graphical terms, |x - 0| = 4 tells us:
4 units
-6
-4
-2
4 units
0
2
4
6
x=4
or
x = -4
© William James Calhoun, 2001
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7-6: Solving Open Sentences Involving Absolute Value
EXAMPLE 1: Solve |x - 3| = 5.
Use the definition of absolute value.
This problem is saying, “The distance between x and 3 is 5 units.”
Graphically:
Run both directions 5 units.
5 units
-3
-1
1
5 units
3
5
7
9
Center on the 3.
The solution is then:
x = -2 or x = 8
{-2, 8}
Another way to solve this problem without graphing it first follows.
The problem tells us that the distance between x and 3 is 5 units, so we know either:
x-3=5
+3 +3
x=8
or
solve
these
equations
x - 3 = -5
+3 +3
x = -2
The solution is then:
x = -2 or x = 8
{-2, 8}
This second method is the CPM and is how I recommend you work these problems.
© William James Calhoun, 2001
7-6: Solving Open Sentences Involving Absolute Value
Now, what if we introduce inequalities into the absolute value mix.
What does |x| < n mean?
Again, this can be rewritten as:
|x - 0| < n
This tells us that, “The distance between x and zero is less than n.”
Using a real number for n, we can see graphically what is going on.
Now, run four units to the left,
Center on the 0.
|x| < 4
but do not include the number
Run four units to the right, but
aka
do not include the number four negative four since it is not “or
equal to.”
since it is not “or equal to.”
|x - 0| < 4
4 units
-6
-4
-2
Since the problem says the
distance is “less than 4,” we
need to shade everything
between the center line and 4.
4 units
0
2
4
Since the problem says the distance
is “less than 4,” we need to shade
everything between the center line
and -4.
6
The answer to the inequality, |x| < 4 is then:
{x | -4 < x < 4}
If |x| < 4 then we can see that either:
x < 4 AND x > -4
We will use this to solve inequalities.
© William James Calhoun, 2001
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7-6: Solving Open Sentences Involving Absolute Value
EXAMPLE 2: Solve |3 + 2x| < 11 and graph the solution set.
Use what we just saw to rewrite the problem as two inequalities connected with “and.”
3 + 2x < 11
-3
-3
2x < 8
2
2
x<4
and
3 + 2x > -11
-3
-3
2x > -14
2
2
x > -7
Now solve the
inequalities.
and
Notice the inequality switch and
sign change on the 11!
{x | -7 < x < 4}
Graph it:
-8
-6
-4
-2
0
2
4
© William James Calhoun, 2001
7-6: Solving Open Sentences Involving Absolute Value
When the problem was an absolute value “less than” something, like
this:
|x - #| < #
the solution had “and” in it because the set of answers were contained
between two numbers.
When the problem is an absolute value “greater than” something, like
this:
|x - #| > #
the solution will be different.
The solution set is not contained by the end number.
The solution set will be outside the bounds of the end number.
The solution will NOT contain “and”, so it must contain…
OR.
© William James Calhoun, 2001
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7-6: Solving Open Sentences Involving Absolute Value
Look at |x| > 4.
Remember this can be written as:
|x - 0| > 4.
This tells us, “The distance between x and zero is greater than 4 units.”
Graphically:
Center on the 0.
Run four units to the right, but
do not include the number four
since it is not “or equal to.”
Since the problem says the
distance is “greater than 4,”
we need to shade everything
to the right of 4.
4 units
-6
-4
-2
Now, run four units to the left,
but do not include the number
negative four since it is not “or
equal to.”
Since the problem says the
distance is “greater than 4,” we
need to shade everything to the
left of -4.
The solution set is:
{x | x < -4 or x > 4}
Out of set-builder
notation:
x > 4 or x < -4
4 units
0
2
6
4
We will solve greater than absolute value inequalities
the same way as lesser than absolute value inequalities.
Just remember:
< yields an and answer, and
> yields an or answer.
© William James Calhoun, 2001
7-6: Solving Open Sentences Involving Absolute Value
EXAMPLE 3: Solve |5 + 2y| ≥ 3 and graph the solution set.
Use what we just saw to rewrite the problem as two inequalities connected with “or.”
5 + 2y ≥ 3
-5
-5
2y ≥ -2
2
2
y ≥ -1
5 + 2y ≤ -3
-5
-5
2y ≤ -8
2
2
y ≤ -4
or
Now solve the
inequalities.
or
Notice the inequality switch and
sign change on the 11!
{y | y ≥ -1 or y ≤ -4}
Graph it:
-8
-6
-4
-2
0
2
4
© William James Calhoun, 2001
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7-6: Solving Open Sentences Involving Absolute Value
HOMEWORK
Page 424
#19 - 37 odd
© William James Calhoun, 2001
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