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Absolute Values

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by Uva@90 » Sun May 24, 2015 5:58 am
What is the value of X ?

1) |X| < 2
2) |X| = 3X-2

OA B

I feel OA is wrong could you please advice me what would be OA ?

Regards,
Uva
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Sun May 24, 2015 6:18 am
Uva@90 wrote:What is the value of x ?

1) |x| < 2
2) |x| = 3x - 2
Target question: What is the value of x?

Statement 1: |x| < 2
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 1 (notice that < 2)
Case b: x = 0 (notice that |0| < 2)
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: |x| = 3x - 2
When solving equation involving ABSOLUTE VALUE, there are 3 steps:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

So, first we get:
x = 3x - 2
Solve, to get x = 1
To check whether this is an extraneous, plug x = 1 into the original equation to get: |1| = 3(1) - 2
Simplify to get: |1| = 1
PERFECT, this solution checks out.

Next, we get:
x = -(3x - 2)
Solve, to get x = 1/2
To check whether this is an extraneous, plug x = 1/2 into the original equation to get: |1/2| = 3(1/2) - 2
Simplify to get: |1/2| = -1/2
This solution DOES NOT check out.

So, it MUST be the case that x = 1
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
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by Ian Stewart » Sun May 24, 2015 6:28 am
Statement 2 is sufficient alone. There are two ways to see that - we have

3x - 2 = |x|
3x = |x| + 2

Notice now that |x| + 2 must be a positive number (since we're adding 2 to something which is zero or greater), so 3x is equal to a positive number, which means x must be positive. If x is positive, then |x| = x, so the equation becomes

3x = x + 2
2x = 2
x = 1

So Statement 2 is sufficient.

I think many test takers will instead use a 'cases' approach, which will also work, as long as you are careful about what you're doing. We start with the equation:

|x| = 3x - 2

If x > 0, then |x| = x, so we get the equation we solved above, and x = 1.

If x < 0, then |x| = -x, and we have

-x = 3x - 2
4x = 2
x = 1/2

But we began by assuming that x was negative, and found that x = 1/2, which is not negative. Our solution contradicts our assumption - it is not a legitimate solution. You can confirm that if you like by plugging it back into the original equation; you'll find it doesn't work.

Since Statement 1 is not sufficient alone, the answer is B.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by Uva@90 » Sun May 24, 2015 9:10 am
Brent@GMATPrepNow wrote:
Uva@90 wrote:What is the value of x ?

1) |x| < 2
2) |x| = 3x - 2
Target question: What is the value of x?

Statement 1: |x| < 2
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 1 (notice that < 2)
Case b: x = 0 (notice that |0| < 2)
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: |x| = 3x - 2
When solving equation involving ABSOLUTE VALUE, there are 3 steps:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

So, first we get:
x = 3x - 2
Solve, to get x = 1
To check whether this is an extraneous, plug x = 1 into the original equation to get: |1| = 3(1) - 2
Simplify to get: |1| = 1
PERFECT, this solution checks out.

Next, we get:
x = -(3x - 2)
Solve, to get x = 1/2
To check whether this is an extraneous, plug x = 1/2 into the original equation to get: |1/2| = 3(1/2) - 2
Simplify to get: |1/2| = -1/2
This solution DOES NOT check out.

So, it MUST be the case that x = 1
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
Thanks Brent,

I missed the third step of the three steps, hence ended in two solution.

When solving equation involving ABSOLUTE VALUE, there are 3 steps:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Thanks again.

Regards,
Uva.
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