NOTE: this is one of those questions that require us to check/test each answer choice. In these situations, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top.

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

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

So, from step 1, we get: $x+2=-x$ and $|x+2=x$
Let's solve each of these independently.

Take: $x+2=-x$
Subtract $x$ from both sides to get: $2=-2x$
Solve: $x=-1$
Plug $x=-1$ into the original equation to get $|(-1)+2|=-(-1)$. Works!!
So, $x=-1$ is one possible solution.


Now take: $x+2=x$
Subtract $x$ from both sides to get: $2=0$
No solution!


So, $x=-1$ is the only solution

Answer: E

Cheers,
Brent

I am not able to eliminate answer choices A and B, could you please explain how those options are not the answer?

Unfortunately, there's no algebraic technique (within the scope of the GRE) to solve equations A, B, and C.
However, if we test a few integer values (from 0 to 4), we'll quickly find some solutions

A. $x|x|=2^x$
x = 2 and x = 4 work here

B. $x+|x|=2^x$
x = 1 and x = 2 work here

C. $2|x|=2^x$
x = 1 and x = 2 work here

Cheers,
Brent

Alright! Thank you

Could you kindly show how the the first four options are solved?

Please see https://gre.myprepclub.com/forum/which-of- ... tml#p51526

I don't see any way to get more than one answer for the equation D. Shouldn't it be an answer too?

The GRE tutor has explained extensively the question above. Please refer to the replies above for a deep understanding.
Regards

Unfortunately I don't see where the respectful tutor explained why D may have more than one 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

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

I see where is my mistake. I didn't check if it satisfies the function itself, thank!