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

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
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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

 

­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.

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.

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