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If \(|x-y|=950\) and \(|y-z|=987\), what is/are the possible value(s) of \(|x-z|?\)

 Oct 24, 2017
 #1
avatar+7350 
+2

| x - y |  =  950

x - y  =  ± 950

x  = ± 950 + y

 

|y - z|  =  987

y - z  =  ± 987

z  =  ±987 + y

 

x - z  =  (± 950 + y) - (± 987 + y )  =  ± 950 + y ± 987 - y

 

x - z  =  ± 950 ± 987

 

x - z = 950 + 987     or     x - z = 950 - 987     or     x - z = -950 + 987     or     x - z = -950 - 987

 

So

 

x - z  =  1937     or     x - z  =  -37     or     x - z  =  37     or     x - z  =  -1937

 

So the possible values for

 

|x - z|   are    1937   and     37     smiley

 Oct 24, 2017
 #2
avatar+98044 
+2

x - y  =   950                         

y - z =    987    add these

 

x - z  =  1937      so  l  x - z l =  1937

 

x - y  =  -950

y - z =    987   add these

 

x - z  =  37       so   l x - z l  =   37

 

The only two other possibilites   are x - y  = -1937    and x - y  = -37......but the absolute values of these are already accounted for by the above answers....

 

 

cool cool cool

 Oct 24, 2017
edited by CPhill  Oct 24, 2017
 #3
avatar+2340 
+2

Doing this problem requires one to consider a few cases. 

 

Firstly, I will solve for x in the first given equation.

 

\(|x-y|=950\) Drop the absolute value bars and split this equation into a positive and negative answer.
\(x-y=950\) \(x-y=-950\)

 

Add to both sides in both cases to isolate x.
\(x_1=y+950\) \(x_2=y-950\)

 

 
   

 

Now, let's solve for z in the second equation in the exact same fashion. 

 

\(|y-z|=987\) Drop the absolute value bars again.
\(y-z=987\) \(y-z=-987\)

 

Subtract y from both sides.
\(-z=-y+987\) \(-z=-y-987\)

 

Divide by -1 to fully isolate.
\(z_1=y-987\) \(z_2=y+987\)

 

 
   

 

In order to solve this problem, one must consider all 4 cases. I have created them all in a table for you! Then, simplify as much as possible.
 

Case 1: \(|x_1-z_1|\) Case 2: \(|x_1-z_2|\) Case 3: \(|x_2-z_1|\) Case 4: \(|x_2-z_2|\)
\(|y+950-(y-987)|\) \(|y+950-(y+987)|\) \(|y-950-(y-987)|\) \(|y-950-(y+987)|\)
\(|y+950-y+987|\) \(|y+950-y-987|\) \(|y-950-y+987|\) \(|y-950-y-987|\)
\(|950+987|\) \(|950-987|\) \(|-950+987|\) \(|-950-987|\)
\(|1937|\) \(|-37|\) \(|37|\) \(|-1937|\)
\(1937\) \(37\) \(37\) \(1937\)
       

 

Therefore, \(|x-z|=37\hspace{1mm}\text{or}\hspace{1mm}|x-z|=1937\)

.
 Oct 24, 2017

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