ash4gmat wrote:Mitch,
Can you also explain one example where we have difference of two absolute values.
If |x+2| - |x-4| = 2, what is the value of x?
|a-b| = the distance between a and b.
|a+b| = |a - (-b)| = the distance between a and -b.
Question stem, rephrased:
|x - (-2)| - |x-4| = 2.
The first absolute value = the distance between x and -2.
The second absolute value = the distance between x and 4.
The greatest possible difference between the two distances is equal to the POSITIVE DIFFERENCE between the two numbers in the absolute values:
4 - (-2) = 6.
The least possible difference between the two distances is equal to the NEGATIVE DIFFERENCE between the two numbers in the absolute values:
-2 - 4 = -6.
The difference between the two distances will be equal to 0 when the two distances are EQUAL -- in other words, when x is HALFWAY between the two numbers in the absolute values.
Halfway between the two numbers = the average of the two numbers = (-2 + 4)/2 = 1.
If x=1, we get:
|x+2| - |x-4| = |1+2| - |1-4| = 3 - 3 = 0.
Here, the number line looks as follows:
-2<-----------x=1----------->4.
Every time x shifts one place toward the right endpoint (4), the first distance will INCREASE by 1, while the second distance will DECREASE by 1.
As a result, the difference between the two distances will INCREASE BY 2.
If x=2, we get:
|x+2| - |x-4| = |2+2| - |2-4| = 4-2 = 2.
If x=3, we get:
|x+2| - |x-4| = |3+2| - |3-4| = 5-1 = 4.
If x=4, we get:
|x+2| - |x-4| = |4+2| - |4-4| = 6-0 = 6.
If x>4, then the difference between the distances will remain 6, since 6 is the maximum possible difference.
If x=5, we get:
|x+2| - |x-4| = |5+2| - |5-4| = 7-1 = 6.
Every time x shifts one place toward the left endpoint (-2), the first distance will DECREASE by 1, while the second distance will INCREASE by 1.
As a result, the difference between the two distances will DECREASE BY 2.
If x=0, we get:
|x+2| - |x-4| = |0+2| - |0-4| = 2-4 = -2.
If x=-1, we get:
|x+2| - |x-4| = |-1+2| - |-1-4| = 1-5 = -4.
If x=-2, we get:
|x+2| - |x-4| = |-2+2| - |-2-4| = 0-6 = -6.
If x<-2, then the difference between the distances will remain -6, since -6 is the least possible difference.
If x=-3, we get:
|x+2| - |x-4| = |-3+2| - |-3-4| = 1-7 = -6.
As illustrated by the portion in red, |x+2| - |x-4| = 2 when x=2.
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