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August 21st, 2018, 10:00 AM   #1
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Dividing by an Absolute Value

Help. I cannot for the life of me find a tutorial on how to divide an integer by an absolute value. I am supposed to simplify this expression:

3 + ( 8 ) รท | 4 |
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August 21st, 2018, 02:04 PM   #2
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As |4| = 4, 3 + 8/4 = 3 + 2 = 5.
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August 21st, 2018, 02:41 PM   #3
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It's seriously that simple? Wow. They made that look unnecessarily complicated. Thanks a bunch!
So it's just the simplified answer in the bars converted into a positive integer?
Is that always the case?
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August 21st, 2018, 04:05 PM   #4
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Quote:
Originally Posted by inomztietuseoe View Post
It's seriously that simple? Wow. They made that look unnecessarily complicated. Thanks a bunch!
So it's just the simplified answer in the bars converted into a positive integer?
Is that always the case?
It is not quite THAT simple.

$x < 0 \implies |x| = -\ x > 0.$

$x \ge 0 \implies |x| = x \ge 0.$

That is the basic idea, but

$|3 - y| = y - 3 \text { if } y > 3 \text { but } |3 - y| = 3 - y \text { if } y \le 3.$

Notice that if y = 3, the absolute value is not a positive integer, but a non-negative integer. More importantly, if you have an expression with a variable within the absolute value bars, you may have to deal with multiple cases.

$|z^2 - 6z - 16| = z^2 - 6z - 16 \text { if } z \le -\ 2.$

$|z^2 - 6z - 16| = 16 + 6z - z^2 \text { if } -\ 2 < z < 8.$

$|z^2 - 6z - 16| = z^2 - 6z - 16 \text { if } z \ge 8.$
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August 21st, 2018, 04:18 PM   #5
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As far as specific real numbers (not necessarily integers) are concerned, the bars have the effect of removing any negative sign, so |4| = |-4| = 4, |1/2| = |-1/2| = 1/2, and |0| = 0.
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