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Director  Affiliations: FRM Charter holder
Joined: 02 Dec 2006
Posts: 605
Schools: Stanford, Chicago Booth, Babson College
If |d-9| = 2d, then d=  [#permalink]

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Difficulty:   15% (low)

Question Stats: 72% (01:09) correct 28% (01:04) wrong based on 1066 sessions

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If |d - 9| = 2d, then d=

(A) -9
(B) -3
(C) 1
(D) 3
(E) 9
Math Expert V
Joined: 02 Sep 2009
Posts: 61385
If |d-9| = 2d, then d=  [#permalink]

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aurobindo wrote:
If |d-9| = 2d, then d=
(A) -9
(B) -3
(C) 1
(D) 3
(E) 9

You can approach this problem in several ways. For example: given |d-9| = 2d --> as LHS (|d-9|) is an absolute value then it's non-negative so RHS (2d or simply d) must also be non-negative thus answer choices A and B are out. Next you can quickly substitute the values to see that d=3 satisfies given inequality: |3-9|=|-6|=6=2*3.

Or you can try algebraic approach and expand |d-9| for 2 ranges:
If $$0\leq{d}\leq{9}$$ then $$-(d-9)=2d$$ --> $$d=3$$ --> you have an answer D right away;
Just to check the second range: If $${d}>9$$ then $$d-9=2d$$ --> $$d=-9$$ --> not a valid solution as $$d$$ cannot be negative (also this value is not in the range we are considering).

Hope it's clear.
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If |d-9| = 2d, then d=
(A) -9
(B) -3
(C) 1
(D) 3
(E) 9

now the two eqs are
1) when (d-9) > 0
d-9 = 2d
d = -9

2) when (d-9) < 0
9-d = 2d
d = 3

the two initial solns are d = -9 and 3

but when we substitute d = -9 in original equation we get
|-9-9| = 2 * -9
18 = -18
which is not possible

Hence only solution is d = 3.
##### General Discussion
Intern  Joined: 11 Dec 2006
Posts: 41

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2
1
if d>9, on solving eqn u get d = -9 which is impossible since d>9.
if d<9, on sloving u get d = 3. Hence D is correct answer
Senior Manager  Joined: 24 Oct 2006
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IMO

2d is an absolute value, so d can't be negative.

Out of the +ve nos. d can be only 3
VP  Joined: 01 May 2006
Posts: 1473

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Sumithra wrote:
IMO

2d is an absolute value, so d can't be negative.

Out of the +ve nos. d can be only 3

Same approach : it's better to plug 2 values mentally with the respect of abs always positive (or 0) than to solve the original equation (saving energy... 4 hours is long) Senior Manager  Status: Math is psycho-logical
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Location: Netherlands
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If |d-9| = 2d, then d=  [#permalink]

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I am having some problems with the range when there is an absolute value. Can I find some good material that explains how we get to the range?

Thank you,
Natalia
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Re: If |d-9| = 2d, then d=  [#permalink]

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Hi All,

Since the answer choices to this question are NUMBERS, we can use them (along with some Number Property knowledge) to quickly get to the solution by TESTing THE ANSWERS.

We're given |D - 9| = 2D and we're asked to solve for D

Since the "left" side of the equation will end up as either a 0 or a POSITIVE, the "right side" of the equation CAN'T be negative, so we know that D CANNOT be NEGATIVE.
Eliminate A and B.

The solution MUST be one of the remaining 3 answers, so we can just TEST them until we find the correct one.

Could D = 1?
|1-9| = |-8| = 8
2D = 2(1) = 2
-8 does NOT = 2
Eliminate C.

Could D = 3?
|3-9| = |-6| = 6
2(3) = 6
6 DOES = 6

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Re: If |d-9| = 2d, then d=  [#permalink]

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pacifist85 wrote:
I am having some problems with the range when there is an absolute value. Can I find some good material that explains how we get to the range?

Thank you,
Natalia

Hi Natalia! (Looks like you are Russian=))
I think you can read in Gmat Club Mathbook if you still need this information
_________________
"Are you gangsters?" - "No we are Russians!"
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3250
If |d-9| = 2d, then d=  [#permalink]

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2
1
Here's a more visual way to think through the given equation |d - 9| = 2d.

|d-9| represents the distance between point d and 9 on the number line. Now, there are only 2 options - either the point d can lie on the LEFT hand side of 9 (At a distance of |d-9| units from 9) or on the RIGHT hand side of 9.

So, let's depict these two cases on the number line. Case 1: d < 9

In this case, |d - 9| = 9 - d (also written as -(d-9))

So, the given equation becomes:

9 - d = 2d
=> d = 3

Case 2: d > 9

In this case, |d - 9| = d - 9

So, the given equation becomes:

d - 9 = 2d
=> d = -9

But this value of d contradicts the condition of Case 2, that d is greater than 9. Therefore, this value of d can be rejected.

So, we get d = 3.

Usually, this visual way of thinking through absolute value expressions helps a lot in situations where you find yourself getting confused about how to open an absolute value expression, what signs to put, what cases to consider etc.

Hope this helped! Japinder
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Re: If |d-9| = 2d, then d=  [#permalink]

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Top Contributor
aurobindo wrote:
If |d - 9| = 2d, then d=

(A) -9
(B) -3
(C) 1
(D) 3
(E) 9

We have two cases to consider:
d - 9 = 2d and d - 9 = -2d

case a: If d - 9 = 2d, then d = -9
When we check this solution for extraneous roots, we get: |-9 - 9| = (2)(-9)
Simplify to get: |-18| = -18
NO GOOD!
So, d = -9 is NOT a valid solution

case b: If d - 9 = -2d, then d = 3
When we check this solution for extraneous roots, we get: |3 - 9| = (2)(3)
Simplify to get: |-6| = 6
WORKS!
So, d = 3 IS a valid solution.

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Re: If |d-9| = 2d, then d=  [#permalink]

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aurobindo wrote:
If |d - 9| = 2d, then d=

(A) -9
(B) -3
(C) 1
(D) 3
(E) 9

First, we know that D has to be a positive, because + x + is always a positive; and the absolute value is always positive. Eliminate A, B
Eliminate 1 because, 8 does not equal 2, eliminate 9 because 0 does not equal 18

Therefore (D) 3 is the answer
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Re: If |d-9| = 2d, then d=  [#permalink]

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aurobindo wrote:
If |d - 9| = 2d, then d=

(A) -9
(B) -3
(C) 1
(D) 3
(E) 9

We can solve the equation, first, when (d - 9) is positive, and second, when (d - 9) is negative.

When (d - 9) is positive:

d - 9 = 2d

-9 = d

Looking at the original equation, we see that if d = -9, then |d - 9| = |-18| = 18, but 2d = 2(-9) = -18. We see that is not possible.

So, let’s now solve the equation when (d - 9) is negative.

-(d - 9) = 2d

-d + 9 = 2d

9 = 3d

3 = d

We see that if d = 3, then |3 - 9| = |-6| = 6 and 2d = 2(3) = 6. Thus, d must be 3.

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Re: If |d-9| = 2d, then d=  [#permalink]

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/d-9/ = will give us one negative and one positive outcome.
D will not take negative as in the equation it equals to a positive 2D , hence D needs a positive value .
Solving the above equation gives answer option D as the only answer.
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Re: If |d-9| = 2d, then d=  [#permalink]

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