If |x + 2| = |x|, how many values of x satisfy this equation?

Question

If  |x + 2| = |x| , how many values of  x  satisfy this equation?

A. 0
B. 1
C. 2
D. 3
E. 4

Bunuel · Accepted Answer

Official Solution:

If  |x + 2| = |x| , how many values of  x  satisfy this equation?

A. 0
B. 1
C. 2
D. 3
E. 4

Square to get rid of the absolute value (note here that we can safely do that since both sides of the equation are non-negative):

x^2+4x+4=x^2 ;

x=-1 .

Therefore, only one value of  x  satisfies the given equation.

Answer: B

gmatophobia · Answer

|x + 2| = |x|

Squaring both sides of the equation

x^2 + 4x + 4 = x^2

4x = -4

x = -1

Hence, only for one value of  x ,  |x + 2| = |x|

Option B

GMATinsight · Answer

Method-1: Just think Logically

Given:  |x + 2| = |x| 
- This is possible only if adding two units to the value of x doesn't change the absolute value
- Which is possible only if the value is turning from negative to positive
- Now, we need to think of two values that are 2 units apart but are the same in absolute values

and I can think of -1 and +1
i.e. x = -1
i.e. one value
Answer: Option B

Method-2: Solve Mathmatically

Given:  |x + 2| = |x| 
i.e. ±(x+2) = ±x
Case 1: +(x+2) = +x i.e. 2 = 0 NOT POSSIBLE
Case 2: +(x+2) = -x i.e. x = -1 First solution
Case 3: -(x+2) = +x i.e. x = 2 NOT POSSIBLE as it doesn't satisfy the primary equation on substituting back
Case 4: -(x+2) = -x i.e. -2 = 0 NOT POSSIBLE

Hence, one solution

Answer: Option B

chetan2u · Answer

You could square to get the answer 
  |x + 2| = |x|......x^2+4x+4=x^2......4x=-4........x=-1

Sometimes, Absolute Modulus questions become very easy when we think of a number line. 
  |x + 2| = |x|  means  |x + 2| = |x+0| , that is x distance from 0 is equal to x distance from -2. Surely x has to be in middle of 0 and -2, which is -1.

Just one value: -1.

B

slykilllrs · Answer

Doesnt case 3 also equal -1? distributing the (-) to inside the perenthesis gives -x-2 = x  --> -2 = 2x --> x = -1

I got the right answer using logical thinking but just wanted to confirm im not missing anything if I had tried to do it mathematically.

chetan2u · Answer

You are absolutely correct.

Rather Case 2 and Case 3 are exactly the same and should have same answer, while Case 1 and case 4 are similar.

Posted from my mobile device

Bunuel · Answer

|a - b| represents the distance between a and b on the number line. For example, the distance between 7 and 17 is |7 - 17| = 10, and the distance between 3 and -2 is |3 - (-2)| = |3 + 2| = 5.

Similarly, |x + 2| represents the distance between x and -2. For example, if x were 10, the distance between 10 and -2 would be |10 - (-2)| = |10 + 2| = 12.

sspiral2556 · Answer

Hi sorry, how do you tell if both sides of the equation are non-negative? Normally, when you open up the modulus, you'd need to account for both positive and negative. How do you instinctively know when it is safe to square both sides right away in this scenario?

Bunuel · Answer

Absolute values are always non-negative by definition. Since both sides of the equation are absolute values (|x+2| and |x|), they can never be negative. That’s why it is safe to square both sides right away, squaring non-negative expressions preserves equality and won’t introduce extraneous solutions.

Also, it’s not correct to say we must always “account for both positive and negative” when opening a modulus. By definition, absolute value outputs a non-negative number in every case. What we really do when we remove the modulus is apply the correct sign (+ or -) to the inside expression so that the result stays non-negative. For example:

If x ≥ 0, then |x| = x. 
 If x < 0, then |x| = -x, and since -x is positive in that case, |x| still comes out non-negative.

Hope it's clear.

If |x + 2| = |x|, how many values of x satisfy this equation?

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If |x + 2| = |x|, how many values of x satisfy this equation?

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