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If |x-6|=2x, then x=?

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Math Revolution GMAT Instructor
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If |x-6|=2x, then x=?  [#permalink]

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New post 25 Oct 2018, 01:22
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[Math Revolution GMAT math practice question]

If \(|x-6|=2x\), then \(x=?\)

\(A. -6\)
\(B. -4\)
\(C. 0\)
\(D. 2\)
\(E. 6\)

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Re: If |x-6|=2x, then x=?  [#permalink]

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New post 25 Oct 2018, 06:04
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Top Contributor
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(|x-6|=2x\), then \(x=?\)

\(A. -6\)
\(B. -4\)
\(C. 0\)
\(D. 2\)
\(E. 6\)


There are 3 steps to solving equations involving ABSOLUTE VALUE:
1. Apply the rule that says: If |x| = k, then x = k or x = -k
2. Solve the resulting equations
3. Plug solutions into original equation to check for extraneous roots

Step 1: x - 6 = 2x or x - 6 = -2x

case a: x - 6 = 2x
Step 2: Solve to get x = -6
Step 3: Plug solution into original equation to get: |(-6) - 6| = 2(-6)
Evaluate to get: |-12| = -12 DOESN'T WORK

case b: x - 6 = -2x
Step 2: Solve to get x = 2
Step 3: Plug solution into original equation to get: |(2) - 6| = 2(2)
Evaluate to get: |-4| = 4 WORK!

Answer: D

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Re: If |x-6|=2x, then x=?  [#permalink]

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New post 25 Oct 2018, 05:28
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(|x-6|=2x\), then \(x=?\)

\(A. -6\)
\(B. -4\)
\(C. 0\)
\(D. 2\)
\(E. 6\)


2x is positive so x is positive...
|x-6|=2x
I...
x-6=2x....x=-6 but x>0, so wrong
II..
x-6=-2x....3x=6...x=2.. correct

D
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If |x-6|=2x, then x=?  [#permalink]

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New post 25 Oct 2018, 07:22
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(|x-6|=2x\), then \(x=?\)

\(A. -6\)
\(B. -4\)
\(C. 0\)
\(D. 2\)
\(E. 6\)


PLUGGIN IN THE OPTIONS

(A) LHS = 12; RHS = -12 (Not equall)
(B) LHS = 10 ; RHS = -8 (Not equall)
(C) LHS = 6 ; RHS = 0 (Not equall)
(D) LHS = 4 RHS = 4 (Possible)
(E) LHS = 0 ; RHS = 12 (Not equall)

Thus, Answer must be (D)
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Re: If |x-6|=2x, then x=?  [#permalink]

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New post 28 Oct 2018, 18:27
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(|x-6|=2x\), then \(x=?\)

\(A. -6\)
\(B. -4\)
\(C. 0\)
\(D. 2\)
\(E. 6\)


As with any absolute value equation, we must consider two cases:

Case 1. We solve for x when (x - 6) is positive:

x - 6 = 2x

-6 = x

Since 2(-6) = -12, and since |x - 6| cannot be -12, then -6 is not a possible value for x.

Case 2. We solve for x when (x - 6) is negative:

-(x - 6) = 2x

-x + 6 = 2x

6 = 3x

2 = x

Answer: D
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Re: If |x-6|=2x, then x=?  [#permalink]

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New post 28 Oct 2018, 19:18
=>

\(|x-6|=2x\)
\(=> x-6 = ±2x\)
\(=> -6 = -x ±2x\)
\(=> -6 = x\) or \(-6 = -3x\)
\(=> x = -6\) or \(x = 2\)
However, \(2x = |x-6| ≥ 0.\)
Thus, we have the unique solution \(x=2.\)


Therefore, the answer is D.
Answer: D
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Re: If |x-6|=2x, then x=?   [#permalink] 28 Oct 2018, 19:18
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