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Intern  B
Joined: 11 Dec 2017
Posts: 6
What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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3 00:00

Difficulty:   85% (hard)

Question Stats: 38% (01:36) correct 62% (01:38) wrong based on 108 sessions

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What is the value of x?

(1) |x + 9| = 2x

(2) |2x − 9| = x
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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4
1
What is the value of x?

(1) |x + 9| = 2x.

The left hand side is an absolute value, so it cannot be negative, thus the right hand side also cannot be negative, which means that x is positive or 0. If $$x \geq 0$$, then x + 9 > 0, thus |x + 9| = x + 9. So, we'd have that x + 9 = 2x. Solving gives x = 9. Sufficient.

(2) |2x − 9| = x.

2x - 9 = x --> x = 9. Plug back to verify this solution: |2*9 - 9| = 9 --> OK;
2x - 9 = -x --> x = 3. Plug back to verify this solution: |2*3 - 9| = 3 --> OK.

Not sufficient.

Note that we cannot use trick we used for (1) for (2): yes, |2x − 9| = x also implies that x must be positive but 2x - 9 could be positive as well as negative for positive x.

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Board of Directors P
Joined: 17 Jul 2014
Posts: 2492
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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2
I don't agree with the OA.
1. 2 options:
x+9=2x.
x=9
or
-x-9=2x
-9=3x
x=-3

1 is not sufficient. So A and D are out.

2. 2 options:
2x-9=x
x=9

or
-2x+9=x
9=3x
x=3

not sufficient. B is out

1+2 => x=9

Intern  B
Joined: 11 Dec 2017
Posts: 6
Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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1
mvictor wrote:
I don't agree with the OA.
1. 2 options:
x+9=2x.
x=9
or
-x-9=2x
-9=3x
x=-3

1 is not sufficient. So A and D are out.

2. 2 options:
2x-9=x
x=9

or
-2x+9=x
9=3x
x=3

not sufficient. B is out

1+2 => x=9

I agree with C too, although my answer was marked wrong and OA was A. Thanks!
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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1
mvictor wrote:
I don't agree with the OA.
1. 2 options:
x+9=2x.
x=9
or
-x-9=2x
-9=3x
x=-3

1 is not sufficient. So A and D are out.

2. 2 options:
2x-9=x
x=9

or
-2x+9=x
9=3x
x=3

not sufficient. B is out

1+2 => x=9

For (1): x = -3 does not satisfy |x + 9| = 2x. You could spot it even without plugin back: if x is negative 2x is also negative and it cannot equal to absolute value of a number, which is non-negative.
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Senior Manager  P
Joined: 15 Oct 2017
Posts: 295
GMAT 1: 560 Q42 V25 GMAT 2: 570 Q43 V27 GMAT 3: 710 Q49 V39 Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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1
A.

Solved it by actually plugging values.

A. This gives x+9 = 2x, x=9.
And, x+9=-2x, x=-3.

By plugging and checking for both values, we get equation stands true for only x=9. Sufficient.

B. This gives 2x-9=x, -9=-x, x=9.
And, 2x-9=-x, -3x=-9, x=3.

By plugging both values we get equation stands true for both the values of 9 and 3. No single answer. Insufficient.

Hence, A.
Board of Directors P
Joined: 17 Jul 2014
Posts: 2492
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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Bunuel wrote:
mvictor wrote:
I don't agree with the OA.
1. 2 options:
x+9=2x.
x=9
or
-x-9=2x
-9=3x
x=-3

1 is not sufficient. So A and D are out.

2. 2 options:
2x-9=x
x=9

or
-2x+9=x
9=3x
x=3

not sufficient. B is out

1+2 => x=9

For (1): x = -3 does not satisfy |x + 9| = 2x. You could spot it even without plugin back: if x is negative 2x is also negative and it cannot equal to absolute value of a number, which is non-negative.

I see...a crucial mistake!
I remember making same mistakes before!
Thank you, Bunuel!
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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lee0706 wrote:
What is the value of x?

(1) |x + 9| = 2x

(2) |2x − 9| = x

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1):
i) x + 9 ≥ 0 or x ≥ -9
| x + 9 | = 2x
⇔ x + 9 = 2x
⇔ x = 9
Since x = 9 satisfies the assumption, we take x = 9 as an answer.

ii) x + 9 < 0 or x < -9
| x + 9 | = 2x
⇔ -( x + 9 ) = 2x
⇔ -x -9 = 2x
⇔ 3x = -9
⇔ x = -3
x = -3 doesn't satisfy the assumption x < -9.
We don'e take x = -3 as an answer.

Since we have a unique answer x = 9, the condition 1) is sufficient.

Condition 2)
i) 2x - 9 ≥ 0 or x ≥ 9/2
|2x − 9| = x
⇔ 2x -9 = x
⇔ x = 9
Since x = 9 satisfies the assumption, we take x = 9 as an answer.

ii) 2x - 9 < 0 or x < 9/2
|2x − 9| = x
⇔ -( 2x -9 ) = x
⇔ -2x + 9 = x
⇔ 3x = 9
⇔ x = 3
Since x = 3 satisfies the assumption, we take x = 3 as an answer.

Thus, we have two solutions x = 3 and x = 9.
Since we don't have a unique solution, the condition 2) is not sufficient, the condition 2) is not sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Joined: 07 Dec 2018
Posts: 146
Location: India
Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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Bunuel wrote:
What is the value of x?

(1) |x + 9| = 2x.

The left hand side is an absolute value, so it cannot be negative, thus the right hand side also cannot be negative, which means that x is positive or 0. If $$x \geq 0$$, then x + 9 > 0, thus |x + 9| = x + 9. So, we'd have that x + 9 = 2x. Solving gives x = 9. Sufficient.

(2) |2x − 9| = x.

2x - 9 = x --> x = 9. Plug back to verify this solution: |2*9 - 9| = 9 --> OK;
2x - 9 = -x --> x = 3. Plug back to verify this solution: |2*3 - 9| = 3 --> OK.

Not sufficient.

Note that we cannot use trick we used for (1) for (2): yes, |2x − 9| = x also implies that x must be positive but 2x - 9 could be positive as well as negative for positive x.

So, Can we say that when there is an addition in modulus, it will take only positive values in cases such as this while when there is subtraction,m both values can be possible? Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x   [#permalink] 05 Aug 2019, 09:47
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