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

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

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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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Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

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

A. -6
B. -4
C. 0
D. 2
E. 6
We can PLUG IN THE ANSWERS, which represent the value of x.
Since an absolute value cannot be equal to a negative value, the correct answer cannot be A or B.

D: x=2
If we plug x=2 into |x-6|=2x, we get:
|2-6| = 2*2
|-4| = 4
4 = 4.
Success!

The correct answer is D.

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Max@Math Revolution 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

Cheers,
Brent

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Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

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

A. -6
B. -4
C. 0
D. 2
E. 6
$$?\,\,\,:\,\,\,\,x\,\,\,{\rm{such}}\,\,{\rm{that}}\,\,\,\,\left| {x - 6} \right| = 2x\,\,\,\,\,\left( {{\rm{Note}}\,\,{\rm{that}}\,\,\,2x \ge 0\,\,,\,\,{\rm{hence}}\,\,x \ge 0} \right)$$
$$\left| {x - 6} \right| = 2x\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{squaring}}} \,\,\,\,\,\,\,{\left( {x - 6} \right)^2} = {\left( {2x} \right)^2}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,3{x^2} + 12x - 36 = 0$$
$$3{x^2} + 12x - 36 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x^2} + 4x - 12 = 0\,\,\,\,\,\,\mathop \Rightarrow \limits_{{\rm{product}}\,\, = \,\, - 12}^{{\rm{sum}}\,\, = \,\, - 4} \,\,\,\,\,x = - 6\,\,\,{\rm{or}}\,\,\,x = 2$$
$${\rm{Testing}}\,\,{\rm{each}}\,\,{\rm{potential}}\,\,{\rm{root}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{original}}\,\,{\rm{equation}}\,\,\,\left( * \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = x = 2\,\,\,$$

(*) Important: as mentioned in the very beginning, -6 doesn´t need to be tested. Someone (correctly) may argue that 2 does not need, too (otherwise no alternative choice could be chosen).
Anyway, we want to remind you that, whenever an equation is squared - or put to any positive even power - roots may be "created" and, because of that, the "final checking" is an important procedure.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Portuguese-speakers :: https://www.gmath.com.br

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Max@Math Revolution 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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scott@targettestprep.com



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=>

|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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Only $149 for 3 month Online Course
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