Nupur.nk wrote:If x is a # such that -2 ≤ x ≤ 2, which of the
following has the largest possible absolute value?

A. 3x-1
B. X2-x
C. 3-x
D. x-3
E. x2+1

|a-b| = the DISTANCE between a and b on the number line.

C: |3-x| = the distance between 3 and x.
D: |x-3| = the distance between x and 3.
Since the distance is the same in each case -- and both answer choices can't be correct -- eliminate C and D.

A: |3x-1| = the distance between 3x and 1.
Since -2≤x≤2, the greatest possible distance between 3x and 1 occurs when x=-2, implying that 3x=-6:
|-6-1| = 7.

B: |x²-x| = the distance between x² and x
Since -2≤x≤2, and the square of a value cannot be negative, 0≤x²≤4.
Thus, the greatest possible distance between x² and x occurs when x=-2, implying that x²=4:
|4 - (-2)| = 6.
Since the maximum possible distance in A is greater than the maximum possible distance in B, eliminate B.

E: |x²+1| = |x² - (-1)| = the distance between x² and -1.
Since -2≤x≤2, and the square of a value cannot be negative, 0≤x²≤4.
Thus, the greatest possible distance between x² and -1 occurs when x=±2, implying that x²=4:
|4 - (-1)| = 5.
Since the maximum possible distance in A is greater than the maximum possible distance in E, eliminate E.

The correct answer is E.