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

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.

Answer:

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ADDITIONAL PRACTICE QUESTION

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

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)

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 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
This IS the answer.

Final Answer:

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

Hi Natalia! (Looks like you are Russian=))
I think you can read in Gmat Club Mathbook if you still need this information

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

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

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.

Answer: D

/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.

given:
|d-9| = 2d,
d - 9 = 2d or d - 9 = -2d
An absolute value cannot be negative, so eliminate A and B.
d - 9 = 2d
-9 = d
No negative values, so solve d - 9 = -2d
-9 = -3d
3 = d
Answer: D

Given that |d - 9| = 2d and we need to find the value of d

Let's solve this using two methods

Method 1: Substitution

Now LHS (Left Hand Side) = Absolute value of a number => it can never be negative
and RHS (Right Hand Side) = 2d
=> negative values of d cannot satisfy this as LHS will be non-negative and RHS will become negative

So, options A and B are out.
Now, let's substitute the value of d from choices C,D,E and see which one satisfies the equation

(C) 1. Put d=1 in |d - 9| = 2d and see if satisfies or not
=> | -1 -9 | = |-10| = 10 \(\neq\) 2*-1 = -2 => NOT POSSIBLE

(D) 3. Put d=3 in |d - 9| = 2d and see if satisfies or not
=> | 3 -9 | = |-6| = 6 = 2*3 = 6 => POSSIBLE

(E) 9. Put d=9 in |d - 9| = 2d and see if satisfies or not
=> | 9 -9 | = |0| = 0 \(\neq\) 2*9 = 18 => NOT POSSIBLE

Method 2: Algebra

|d - 9| = 2d
=> d-9 = 2d or d-9 = -2d
=> 2d-d = -9 or d+2d = 9
=> d = -9 or 3d = 9 or d = \(\frac{9}{3}\) = 3
=> d = -9 or d = 3

If we substitute these values of d back in |d-9| = 2d then we will see that d=-9 does not satisfy the equation
=> d = 3

So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

The simplest method would be to plug in the options here.

But if one were to do it algebraically, we would remove the absolute value sign using the definition of absolute value:
|d - 9| = (d - 9) if d >= 9
|d - 9| = - (d - 9) if d < 9

Case 1: d >= 9
(d - 9) = 2d
d = -9
Not valid because d must be >= 9

Case 2: d < 9
-(d - 9) = 2d
d = 3
Valid because here d is less than 9.

Answer (D)

Check this post for more on this concept:
https://anaprep.com/algebra-the-why-beh ... questions/

d-9 = 2d

Or 9-d =2d

D=-9 or d=3

But d can only be positive

Therefore d =3

Therefore D