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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)

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).

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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.

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 lotin situations where you find yourself getting confused about how to open an absolute value expression, what signs to put, what cases to consider etc.

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