|a-b| = the distance between a and b
|a+b| = |a-(-b)| = the distance between a and -b
VJesus12 wrote:If y = |2 + x| - |2 – x| and |2x – 15| < 2, how many integer values can y take?
(A) 0
(B) 1
(C) 2
(D) 4
(E) Cannot be determined
y = |2 + x| - |2 – x|
y = |x+2| - |x-2|
y = |x-(-2)| - |x-2|
y = the subtraction of two distances:
(distance between x and -2) - (distance between x and 2)
On the number line:
..........-2..........2..........
Case 1: x≥2
In this case, y=4 (the distance between the -2 and 2):
x=2 --> y = |2+2| - |2-2| = 4
x=2.1 --> y = |2.1+2| - |2.1-2| = 4
x=10 --> y = |10+2| - |10-2| = 4
Case 2: x≤-2
In this case, y=-4 (the opposite of the distance between the -2 and 2):
x=-2 --> y = |-2+2| - |-2-2| = -4
x=-2.1 --> y = |-2.1+2| - |-2.1-2| = -4
x=-10 --> y = |-10+2| - |-10-2| = -4
Case 3: -2 < x < 2
In this case, -4 < y < 4:
x=1 --> y = |1+2| - |1-2| = 2
x=1.9 --> y = |1.9+2| - |1.9-2| = 3.8
x=-1 --> y = |-1+2| - |-1-2| = -2
Given condition for x:
|2x - 15| < 2
Here, the distance between 2x and 15 must be less than 2.
Thus, 2x can be up to places to the left or right of 15 on the number line:
13<---2 places--->15<---2 places--->17
In other words, 2x must be BETWEEN 13 AND 17:
13 < 2x < 17
6.5 < x < 8.5
Since only Case 1 is applicable, y=4.
The correct answer is
B.
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