Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +
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23 Sep 2017, 13:14
OFFICIAL SOLUTION FROM MANHATTAN
1) SUFFICIENT: It’s impractical to take an algebraic approach to this statement; doing so would entail a large number of cases. For instance, |x – 3| is equal to 3 – x if x < 3, but is equal to x – 3 if x > 3; similarly, the other three absolute-value expressions switch at x = 4, 6, and 7, respectively.
Instead, it’s more efficient to consider the first three positive odd integers (1, 3, and 5) individually and then to consider only one algebraic case, the case in which x > 7 (because when x > 7, all values are positive so we can ignore the absolute value symbols).
If x = 1, then the value is (2) – 2(3) + 2(5) – (6) = 0.
If x = 3, then the value is (0) – 2(1) + 2(3) – (4) = 0.
If x = 5, then the value is (2) – 2(1) + 2(1) – (2) = 0.
If x > 7, then drop the absolute value symbols and simplify:
(x – 3) – 2(x – 4) + 2(x – 6) – (x – 7) =
x – 3 – 2x + 8 + 2x – 12 – x + 7 =
(x – 2x + 2x– x) + (-3 + 8 – 12 + 7) =
0
Therefore, the value of the expression is also 0 for all values of x greater than or equal to 7, including the odd integers 7, 9, 11, and so on. The value of the expression is thus 0 for all positive odd integers. The statement is sufficient.
(2) NOT SUFFICIENT:
As determined during the discussion of statement 1, the expression is equal to 0 when x > 7 (whether odd integer, even integer, or non-integer). We still need to test the non-integer values of x between 6 and 7.
If x = 6.5, then the value is (3.5) – 2(2.5) + 2(0.5) – 0.5 = –1, which is not equal to 0. The expression can thus have multiple values, so the statement is insufficient.
The correct answer is A.