[Math Revolution GMAT math practice question]
If |x-6|=2x, then x=?
A. -6
B. -4
C. 0
D. 2
E. 6
If |x-6|=2x, then x=?
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- Max@Math Revolution
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We can PLUG IN THE ANSWERS, which represent the value of x.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
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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There are 3 steps to solving equations involving ABSOLUTE VALUE: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
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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- fskilnik@GMATH
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$$?\,\,\,:\,\,\,\,x\,\,\,{\rm{such}}\,\,{\rm{that}}\,\,\,\,\left| {x - 6} \right| = 2x\,\,\,\,\,\left( {{\rm{Note}}\,\,{\rm{that}}\,\,\,2x \ge 0\,\,,\,\,{\rm{hence}}\,\,x \ge 0} \right)$$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
$$\left| {x - 6} \right| = 2x\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{squaring}}} \,\,\,\,\,\,\,{\left( {x - 6} \right)^2} = {\left( {2x} \right)^2}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,3{x^2} + 12x - 36 = 0$$
$$3{x^2} + 12x - 36 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x^2} + 4x - 12 = 0\,\,\,\,\,\,\mathop \Rightarrow \limits_{{\rm{product}}\,\, = \,\, - 12}^{{\rm{sum}}\,\, = \,\, - 4} \,\,\,\,\,x = - 6\,\,\,{\rm{or}}\,\,\,x = 2$$
$${\rm{Testing}}\,\,{\rm{each}}\,\,{\rm{potential}}\,\,{\rm{root}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{original}}\,\,{\rm{equation}}\,\,\,\left( * \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = x = 2\,\,\,$$
(*) 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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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As with any absolute value equation, we must consider two cases: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
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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- Max@Math Revolution
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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
|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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