Many absolute value problems can be solved as DISTANCE problems.
|x-y|= the distance between x and y.
|x+y| = |x-(-y)| = the distance between x and -y.
Thus, |x + 1| + |x - 3| = 6 implies the following:
(the distance between x and -1) + (the distance between x and 3) = 6.
In other words, the SUM OF THE TWO DISTANCES = 6.
-1--------x----------3
The distance between -1 and 3 is 4.
If x is BETWEEN -1 and 3, the sum of the two distances will be 4.
For the sum of the two distances to be 6, x must be to the left of the lower endpoint (-1) or to the right of the upper endpoint (3).
For every unit x moves beyond either endpoint, the SUM of the two distances will increase by TWO UNITS.
The reason is that EACH DISTANCE is affected by the movement of x.
Since there are two distances, the effect is TWICE AS GREAT.
Thus:
To increase the sum of the two distances by k units, x must be (1/2)k units to the left of the lower endpoint or (1/2)k units to the right of the upper endpoint.
Since the sum here must increase by 2 units, x must be 1 unit to the left of -1 or 1 unit to the right of 3:
x=-2<----(-1)---------------(3)
----->x=4
Thus, there are two valid solutions for x:
x=-2 and x=4.
Here's another example:
|x-2| + |x+3| = 13.
-3---------------2
The distance between -3 and 2 is 5.
To increase the sum of the two red distances by 8 to 13, x must be 4 units to the left of -3 or 4 units to the right of 2:
x=-7<----(-3)---------------(2)
----->x=6
Thus, there are two valid solutions for x:
x=-7 and x=6.
infiniti007 wrote:What is the value of integer x?
1.) |1-x| - |x+1| = 0
2.) |7-x| + |3-x| = 10
Statement 1: |1-x| - |x+1| = 0
|1-x| = |x+1|
The distance between 1 and x is equal to the distance between x and -1.
In other words, x is HALFWAY between 1 and -1.
Thus, x=0.
SUFFICIENT.
Statement 2: |7-x| + |3-x| = 10
Rephrased:
|x-7| + |x-3| = 10.
3---------------7
The distance between 3 and 7 is 4.
To increase the sum of the two red distances by 6 to 10, x must be 3 units to the left of 3 or 3 units to the right of 7:
x=0<----(3)---------------(2)
----->x=10
Thus, there are two valid solutions for x:
x=0 and x=10.
INSUFFICIENT.
The correct answer is
A.
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