If \(|x-y|=950\) and \(|y-z|=987\), what is/are the possible value(s) of \(|x-z|?\)
+9481 | 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 
+129907 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....
+2446 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. | ||
| Add y to both sides in both cases to isolate x. | ||
| |||
Now, let's solve for z in the second equation in the exact same fashion.
| \(|y-z|=987\) | Drop the absolute value bars again. | ||
| Subtract y from both sides. | ||
| Divide by -1 to fully isolate. | ||
| |||
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, \(|x-z|=37\hspace{1mm}\text{or}\hspace{1mm}|x-z|=1937\)
