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MathGroup Archive 2004

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Re: New user: Abs[] problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50107] Re: [mg50079] New user: Abs[] problem
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Sun, 15 Aug 2004 03:14:38 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200408140550.BAA15327@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

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?

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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