manhattan practice examination #1 non-adaptive

[reshma_menghani]

If x is not equal to 0, is |x| less than 1?

(1) x/|x| < x

(2) |x| > x


Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

The question "Is |x| less than 1?" can be rephrased in the following way.

Case 1: If x > 0, then |x| = x. For instance, |5| = 5. So, if x > 0, then the question becomes "Is x less than 1?"

Case 2: If x < 0, then |x| = -x. For instance, |-5| = -(-5) = 5. So, if x < 0, then the question becomes "Is -x less than 1?" This can be written as follows:

-x < 1?
or, by multiplying both sides by -1, we get
x > -1?

Putting these two cases together, we get the fully rephrased question:
"Is -1 < x < 1 (and x not equal to 0)"?

Another way to achieve this rephrasing is to interpret absolute value as distance from zero on the number line. Asking "Is |x| less than 1?" can then be reinterpreted as "Is x less than 1 unit away from zero on the number line?" or "Is -1 < x < 1?" (The fact that x does not equal zero is given in the question stem.)
(1) INSUFFICIENT: If x > 0, this statement tells us that x > x/x or x > 1. If x < 0, this
statement tells us that x > x/-x or x > -1. This is not enough to tell us if -1 < x < 1.

(2) INSUFFICIENT: When x > 0, x > x which is not true (so x < 0). When x < 0, -x > x or
x < 0. Statement (2) simply tells us that x is negative. This is not enough to tell us if -1 < x < 1.

(1) AND (2) SUFFICIENT: If we know x < 0 (statement 2), we know that x > -1 (statement 1). This
means that -1 < x < 0. This means that x is definitely between -1 and 1.

The correct answer is C.

I think it should have been answer a. You can reduce choice a to
1<|x|. simply divide x on both sides. and then you should be able to get that. so
why can't it be a.

[S]

You can't divide by x like this, since you don't know the sign of x ! If, instead of a variable x, you had a constant like 2 or -4, you'd have known whether or not to flip the inequality direction.

.

If x is not equal to 0, is |x| less than 1?

(1) x/|x| < x

(2) |x| > x


Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

The question "Is |x| less than 1?" can be rephrased in the following way.

Case 1: If x > 0, then |x| = x. For instance, |5| = 5. So, if x > 0, then the question becomes "Is x less than 1?"

Case 2: If x < 0, then |x| = -x. For instance, |-5| = -(-5) = 5. So, if x < 0, then the question becomes "Is -x less than 1?" This can be written as follows:

-x < 1?
or, by multiplying both sides by -1, we get
x > -1?

Putting these two cases together, we get the fully rephrased question:
"Is -1 < x < 1 (and x not equal to 0)"?

Another way to achieve this rephrasing is to interpret absolute value as distance from zero on the number line. Asking "Is |x| less than 1?" can then be reinterpreted as "Is x less than 1 unit away from zero on the number line?" or "Is -1 < x < 1?" (The fact that x does not equal zero is given in the question stem.)
(1) INSUFFICIENT: If x > 0, this statement tells us that x > x/x or x > 1. If x < 0, this
statement tells us that x > x/-x or x > -1. This is not enough to tell us if -1 < x < 1.

(2) INSUFFICIENT: When x > 0, x > x which is not true (so x < 0). When x < 0, -x > x or
x < 0. Statement (2) simply tells us that x is negative. This is not enough to tell us if -1 < x < 1.

(1) AND (2) SUFFICIENT: If we know x < 0 (statement 2), we know that x > -1 (statement 1). This
means that -1 < x < 0. This means that x is definitely between -1 and 1.

The correct answer is C.

I think it should have been answer a. You can reduce choice a to
1<|x|. simply divide x on both sides. and then you should be able to get that. so
why can't it be a.

[RonPurewal]

Very important: You cannot divide an INEQUALITY by a variable. As 'S' has implied above, inequality signs are flipped upon division by a negative number, but NOT upon division by a positive number. Since we don't know whether x is positive or negative, dividing by it produces an awkward situation in which we simply don't know whether to flip the sign. That's bad.

Incidentally, you can prove to yourself that (1) isn't sufficient by considering two cases: x = -1/2 and x = 2. The first of these is between -1 and 1 and the second isn't, so this statement isn't sufficient. (Notice that I've cherry-picked these two cases, which I certainly wouldn't have expected you to have selected right away - BUT, given that you know you're looking for -1 < x < 1, you would be well advised to try numbers both inside and outside this range.)

[Nov1907]

We can multiply by |x| however since it is always positive. Right ? That should reduce the amount of time needed to simplify the problem.

[RonPurewal]

You can, indeed, multiply both sides of an expression by an absolute value. However, in this problem, doing so would lead to an expression of comparable complexity to the one that's already there.