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What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x

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

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19 Feb 2018, 09:55
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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
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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19 Feb 2018, 10:22
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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What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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19 Feb 2018, 10:13
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
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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19 Feb 2018, 10:16
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!
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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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19 Feb 2018, 10:24
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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Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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19 Feb 2018, 10:41
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.
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Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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19 Feb 2018, 11:04
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!
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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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19 Feb 2018, 18:11
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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Re: What is the value of x? (1) |x + 9| = 2x (2) |2x − 9| = x  [#permalink]

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05 Aug 2019, 09:47
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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