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#1 2012-10-27 14:16:16

Sabbir
Guest

Absolute value equation

How do I solve this equation

#2 2012-10-27 15:32:48

noelevans
Full Member

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Re: Absolute value equation

Hi!

The general approach to absolute value equalities is to consider where the quantities inside the
absolute values are positive, zero, or negative.  Try the following:

Where both of 2x+1 and 2x-1 are positive replace the absolute values with these and solve.
Where both of 2x+1 and 2x-1 are negative replace the absolute values with the opposite of these
    and solve.
Where 2x+1 is positive and 2x-1 is negative replace |2x+1| with 2x+1 and |2x-1| with -(2x-1)
    and solve.
You will also find solutions where each of these quantities individually are zero.

You will find the solutions in [-1/2, 1/2]. smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

#3 2012-10-27 16:35:08

Sabbir
Guest

Re: Absolute value equation

Could you please show the solving?

#4 2012-10-27 19:00:01

bob bundy
Moderator

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Re: Absolute value equation

hi Sabbir,

Welcome to the forum.

For any 'linear' expression



The critical value for x is







So, for your equation you need to consider these cases:

(i) x < - 1/2

expression becomes





Although, strictly that value is outside the range that I'm checking, a numeric check shows it is a solution.

(ii) x = -1/2 Just done that.

(iii) -1/2 < x < 1/2

expression becomes



This is true for all values of x in the range.

(iv) x = +1/2

expression becomes



So this is a solution.

(v)

expression becomes





Putting (i) to (v) together, the solution set is {x : -1/2 ≤ x ≤ 1/2}  or [-1/2,1/2] as noelevans has already explained.

Hope that helps,

Bob


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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#5 2012-10-27 22:02:37

anonimnystefy
Real Member

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Re: Absolute value equation

Hi Bob

The solution is correct, but the interval is wrong. It should be an open interval: (-1/2,1/2)=]-1/2,1/2[.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#6 2012-10-27 22:03:58

bobbym
Administrator

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Re: Absolute value equation

Hi;

- 1 / 2 is a solution, isn't it?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#7 2012-10-27 22:05:47

anonimnystefy
Real Member

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Re: Absolute value equation

True. Then the solution set should be {x: -1/2<=x<=1/2}


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#8 2012-10-27 22:43:17

bob bundy
Moderator

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Re: Absolute value equation

Thanks.  In my head it said that, but, as you know, I'm getting old and doddery, so what my fingers type doesn't always square with what my brain thinks.  I have edited the post.

If you look carefully at the graph, you see the end point pixels are included.  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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