SOLUTION: Simplify |x-1| + |x-2| given that x is in the open interval (1,2). Note: The book tells me that since x > 1, then x - 1 is positive. So, |x-1| = x - 1. Where does it say that x

Algebra ->  Absolute-value -> SOLUTION: Simplify |x-1| + |x-2| given that x is in the open interval (1,2). Note: The book tells me that since x > 1, then x - 1 is positive. So, |x-1| = x - 1. Where does it say that x      Log On


   

Question 1207369: Simplify |x-1| + |x-2| given that x is in the open interval (1,2).
Note:
The book tells me that since x > 1, then x - 1 is positive.
So, |x-1| = x - 1.
Where does it say that x > 1?
The book goes on to say that x < 2.
Since x < 2, then x - 2 is negative.
So, |x-2| = -(x-2) = -x + 2.
Where does it say that x < 2?
The author went in to do this:
x - 1 - x + 2 = 1.
The answer is 1.
However, my two questions are:
1. Where does it say that x > 1?
2. Where does it say that x < 2?




Found 3 solutions by ikleyn, Edwin McCravy, mccravyedwin:
Answer by ikleyn(52890) About Me  (Show Source):
You can put this solution on YOUR website!
. 

The problem says that x is in the open interval (1,2).


It means that x > 1       (since on the number line, x is on the right of 1).


It also means that x < 2  (since on the number line, x is on the left of 2).


These are the answers to your questions (1) and (2).

Answered and explained, in full.

Your problem is that you do not understand elementary standard mathematical writing,
in no one of its words, terms and/or conceptions, as well as in no one single mathematical symbol.

It means that you do not have necessary pre-requisite knowledge to read, understand and solve these problems.


To learn Math this way without following right logic is a deliberately losing strategy, which will not lead to success.

To learn mathematics successfully, you need to follow carefully designed courses where all concepts are introduced sequentially.

Good school courses have been developed over centuries, so the best thing you can do is to follow those well-designed courses.



Answer by Edwin McCravy(20064) About Me  (Show Source):
Answer by mccravyedwin(409) About Me  (Show Source):
You can put this solution on YOUR website!
Yes, my intervals were typed wrong.  When I went back to type them in,
I inadvertently used -1 instead of 1.  Here it is corrected below.  It is a
shame that there is no way for tutors to communicate with each other on this
site.  


Tutors, for Pete's sake, TEACH students, don't BLAST them for not
understanding something!!!

(1,2) in this problem is represented by

-----------------o===o----------
-3  -2  -1   0   1   2   3   4

or 

-----------------(===)----------
-3  -2  -1   0   1   2   3   4  


I can understand perfectly well why he/she doesn't understand (1,2). 

Mathematics, unfortunately, has evolved so that (1,2) can mean two entirely
different things:

1.  (1,2) can mean an ordered pair represented by a point with x-coordinate 1
and y-coordinate 2.  It DOES NOT mean that in this problem.

2.  (1,2) can also mean the open interval {x | 1 < x < 2}, which is what it
means in this problem.

Notice the following notation for intervals:

[1,2) means the half-open-half-closed interval {x | 1 < x < 2}.
It means x CAN be 1 or greater than 1 but less than 2, and it CANNOT be 2.

-----------------[===)----------
-3  -2  -1   0   1   2   3   4

****************************************************************

(1,2] means the half-open-half-closed interval {x | 1 < x < 2}.
It means that x CANNOT be 1 but must be greater than 1 but less than 2, and
it CAN be 2.

-----------------(===]----------
-3  -2  -1   0   1   2   3   4

****************************************************************

[1,2] means the closed interval {x | 1 < x < 2}.

-----------------[===]----------
-3  -2  -1   0   1   2   3   4

****************************************************************

It means that x CAN BE 1 or greater that 1 but less than or equal 2.  It CAN
be either 1 or 2 or anything between.

(1,2) means the open interval {x | 1 < x < 2}.  <--THE AMBIGUOUS NOTATION!
It means x must be greater than 1 but not equal to 1, and less than 2 but not
equal to 2.

-----------------(===)----------
-3  -2  -1   0   1   2   3   4

****************************************************************

That is: 

[ on the left means the interval INCLUDES the number on the left.
( on the left means the interval DOES NOT INCLUDE the number on the left.
] on the right means the interval INCLUDES the number on the right.
) on the right means the interval DOES NOT INCLUDE the number on the right.

So always go by the context if you see (1,2) or something similar.  Ask
yourself:  Is the problem about plotting points on a set of x and y coordinates,
or is it about values and intervals on a number line?

So for your problem:

(1,2) tells you that since x > 1, then x - 1 > 0, so x - 1 is positive.

So, |x-1| = x - 1.

(1,2) also tells you that x < 2.

Since x < 2, then x - 2 < 0 or x - 2 is negative.

So, |x-2| = -(x-2) = -x + 2.

(1,2) tells you that x < 2?

Then

(x - 1) + (-x + 2) = x - 1 - x + 2 = 1

The answer is 1.  

Hope this helps. 

Edwin