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Interesting Question, perhaps...

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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
+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

Oct 24, 2017
#2
+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....

Oct 24, 2017
edited by CPhill  Oct 24, 2017
#3
+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, $$|x-z|=37\hspace{1mm}\text{or}\hspace{1mm}|x-z|=1937$$

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Oct 24, 2017