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MathGroup Archive 2004

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Re: Re: New user: Abs[] problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50122] Re: [mg50107] Re: [mg50079] New user: Abs[] problem
  • From: DrBob <drbob at bigfoot.com>
  • Date: Mon, 16 Aug 2004 06:45:28 -0400 (EDT)
  • References: <200408140550.BAA15327@smc.vnet.net> <200408150714.DAA12689@smc.vnet.net>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

You miscopied the problem (I recommend copy/paste). The OP's problem DOES have solutions.

Bobby

On Sun, 15 Aug 2004 03:14:38 -0400 (EDT), Murray Eisenberg <murray at math.umass.edu> wrote:

> Yes, sometimes Mathematica cannot find (all) solutions.  But in this
> case, BELIEVE!  Mathematica DID solve it: there are no solutions!
>
> You can see this, e.g., by graphing your piecewise-linear function:
>
>    Plot[x + 1 + Abs[2x - 4] + Abs[5 - x], {x, -10, 10}];
>    Plot[x + 1 + Abs[2x - 4] + Abs[5 - x], {x, 0, 5}];
>
> (The second Plot is just to convince you that the minimum really is
> strictly positive.)
>
> Or, you can consider the 4 cases:
>
>    case1 = x >= 2 && 5 >= x;
>    case2 = x >= 2 && 5 < x;
>    case3 = x < 2 && 5 >= x;
>    case4 = x < 2 && 5 < x;
>
> The first case gives the closed interval from 2 to 5; the second and
> third simplify...
>
>    Reduce[case2, x]
> x>5
>    Reduce[case3, x]
> x<2
>
> ... and the fourth case gives the empty set:
>
>    Reduce[case4, x]
> False
>
> (You could have done that much without Mahematica, of course.)
> Now look at your function of x on each of these intervals:
>
>    expr = x + 1 + Abs[2x - 4] + Abs[5 - x];
>
>    lin1 = Simplify[expr, case1]
> 2(1+x)
>    lin2 = Simplify[expr, case2]
> 4 (-2 + x)
>    lin3 = Simplify[expr, case3]
> -2 (-5 + x)
>
> Now try to solve the resulting linear equation in each case:
>
>    Solve[lin1 == 0, x]
> {{x->-1}}
>    Solve[lin2 == 0, x]
> {{x->2}}
>    Solve[lin3 == 0, x]
> {{x->5}}
>
> In each case, the line meets the x-axis at a point outside the interval
> on which the corresponding line segment over the interval in question is
> the domain of the piece.  QED
>
>
> Edson.Brusque at smc.vnet.net wrote:
>> Hello,
>>
>>      I'm starting Electrical Engineering and using Mathematica for
>> helping me in Calculus e Algebra studies.
>>
>>      I'm trying to solve this equation on Mathematica:
>>          x + 1 - |2x - 4| + |5 - x| = 0
>>
>>      on the Math* notebook I'm typing:
>>          Solve[x + 1 - Abs[2x - 4] + Abs[5 - x] == 0,x]
>>
>>      but only got an empty output: {{}}
>>
>>      Someone please can help me get Math* to solve this?
>



-- 
DrBob at bigfoot.com
www.eclecticdreams.net


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