Data sufficiency with modulus

[uniyal01]

1---What is the value of |x−2|?
(1) |x−4|=2
(2) |2−x|=4

2---what is the value of x?
(1) |x|=-x
(2) |x|^2 = x^2

Thanks!

[GMATGuruNY]

Please post each problem in a separate thread in the DS forum.

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Hi uniyal01,

In the first DS prompt, we're asked for the value of |X-2|.

1) |X-4| = 2

Since we're dealing with an equation and an absolute value, there will almost certainly be two solutions to this equation.

When X=2.... |2-4| = |-2| = 2
When X=6... |6-4| = |2| = 2

IF....
X=2, then the answer to the question is |2-2| = |0| = 0
X=6, then the answer to the question is |6-2| = |4| = 4
Fact 1 is INSUFFICIENT

2) |2-X| = 4

When X=-2.... |2-(-2)| = |4| = 4
When X=6... |2-6| = |-4| = 4

IF....
X=-2, then the answer to the question is |-2-2| = |-4| = 4
X=6, then the answer to the question is |6-2| = |4| = 4
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

First one:

S1:

Two options:

If (x - 4) ≥ 0, then |x - 4| = x - 4, so x - 4 = 2, and x = 6.

If (x - 4) < 0, then |x - 4| = -(x - 4), so 4 - x = 2, and x = 2.

So |x - 2| could be 4 or could be 0.

S2:

Same idea, two options:

If |2 - x| ≥ 0, then |2 - x| = 2 - x, and 2 - x = 4, and x = -2.

If |2 - x| < 0, then |2 - x| = -(2 - x), and -(2 - x) = 4, and x = 6.

So |x - 2| could be 4 or could be 4. Since these come out the same, S2 is SUFFICIENT.

[Matt@VeritasPrep]

Second one:

S1: |x| = -x

This is a funny way of saying x ≤ 0. So x could be 0, -1, -2, whatever.

S2: |x|² = x²

This is true for any value of x, so it doesn't help at all.

Together we still have x ≤ 0, so we can't solve for one specific value.