sanjoo wrote:
If x is an integer and |1 − x| < 2 then which of the following must be true?
(A) x is not a prime number
(B) x^2 + x is not a prime number
(C) x is positive
(D) Number of distinct positive factors of x + 2 is a prime number
(E) x is not a multiple of an odd prime number
Two properties involving absolute value inequalities:
Property #1: If |something| < k, then –k < something < k
Property #2: If |something| > k, then EITHER something > k OR something < -k Note: these rules assume that k is positiveSince the given inequality, |1 − x| < 2, is in the form |something| < k, we know we need to apply Property #1
When we apply Property #1, we get:
-2 < 1 - x < 2Subtract 1 from all sides to get:
-3 < -x < 1Multiply all sides by -1 to get:
3 > x > -1 [ since we multiplied all sides of the inequality by a negative value, we reversed the direction of the inequality symbols]If x is an INTEGER and if
3 > x > -1, then there are only three possible values of x:
0, 1 or 2Now let's examine the five answer choices:
(A) x is not a prime number
Since x COULD equal
2, x COULD be prime.
(B) x^2 + x is not a prime number
If x =
1, then x^2 + x =
1^2 +
1 = 2, and 2 IS prime.
(C) x is positive
If x =
0, then x is NOT positive.
(D) Number of distinct positive factors of x + 2 is a prime number
If x =
0, then x + 2 = 2. 2 has TWO positive factors (1 and 2), and TWO IS a prime number.
If x =
1, then x + 2 = 3. 3 has TWO positive factors (1 and 3), and TWO IS a prime number.
If x =
2, then x + 2 = 4. 4 has THREE positive factors (1, 2 and 4), and THREE IS a prime number.
(E) x is not a multiple of an odd prime number
If x =
0, then x IS a multiple of an odd prime number
Answer: D