Which of the following has exactly one solution ?

Question

A.  x|x|=2^x

B.  x+|x|=2^x

C.  2|x|=2^x

D.  2|x|=2x-1

E.  |x+2|=-x

GMATinsight · Answer

A.  x|x|=2^x  True for x = 2 and 4 ELIMINATED

B.  x+|x|=2^x  True for x = 1 and 2

C.  2|x|=2^x  True for x = 2 and 1 Eliminated

D.  2|x|=2x-1  NO SOLUTION (This can NOT be true for any positive value of x and negative value in turn gives us positive value 1/4 NOT Acceptable)

E.  |x+2|=-x  x=-1

Answer: Option E

skhemani · Answer

How is x=1 not a solution for C?

Bunuel · Answer

It is. Both x = 1 and x = 2 satisfy 2|x| = 2^x.

kris19 · Answer

C. 2|x| = 2^x  
case (i) x is -ve, then
  |x| = x --> 2*x = \frac{1}{2^{x}}  , which is not possible for any x
case (ii) x is +ve, then
  |x| = x --> 2*x = 2^x  
above equation holds good for x = 1 (2*1 = 2^1) and x = 2 (2*2 = 2^2), so option C has two solutions for x, so not correct

E. |x+2| = -x
case (i) x is -ve, then take 
  x = -1 --> |-1+2| = -(-1)  
  1 = 1  , so x takes -1 ( one solution)
  x= -2 --> |-2+2| = -(-2)  
  0 = 2  , so x can't take -2
  x = -3 --> |-3+2| = -(-3)   so x can't take -3, and so for other negative numbers

case (ii) x is +ve, then
  |x+2| = -x   becomes x+2 = -x --> x = -1
so, x can take -1, from case (i) & (ii), x has only one solution (-1)

so option E is correct

ScottTargetTestPrep · Answer

Solution:

We can solve the equations starting with D since D is the easiest equation of the ones given.

D. 2|x| = 2x - 1

If x is positive, we have 2x = 2x - 1, which yields no solution.

If x is negative, we have -2x = 2x - 1. Solving it, we get:

-4x = -1

x = -1/-4 = 1/4

Since we assumed x was negative, the positive value x = 1/4 may not satisfy the equation. Indeed, if we substitute x = 1/4, we get:

2|1/4| ≟ 2(1/4) - 1

2 * 1/4 ≟ 1/2 - 1

1/2 ≟ -1/2

Since 1/2 is not equal to -1/2, x = 1/4 is not a solution for 2|x| = 2x - 1. Thus, this equation has no solutions.

Next, let’s solve the equation in answer choice E since it is similarly easy to solve.

E. |x + 2| = -x

If (x + 2) is positive, then:

x + 2 = -x

2x = -2

x = -1

We can verify that x = -1 is indeed a solution for |x + 2| = -x.

If (x + 2) is negative, then:

-x - 2 = -x

x + 2 = x

No value of x will satisfy the equation above. Hence, the equation |x + 2| = -x has only one solution.

Answer: E

Sevil92 · Answer

Is this an official question? The rule is that absolute value cannot give us a negative result but here we take it as the correct answer. How can |1| be equal to -1?

Bunuel · Answer

|x + 2| = −x has one solution x = -1:

|-1 + 2| = −(-1);

|1| = 1;

1 = 1.

bumpbot · Answer

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Which of the following has exactly one solution ?

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