What is x ? (1) x^2 - 1 = x + 1 (2) x + 3 is a prime number

Statement One Alone:

x^2 - 1 = x + 1

x^2 - 1 = x + 1

x^2 - x - 2 = 0

(x - 2)(x + 1) = 0

x = 2 or x = -1

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x + 3 is a prime number

x could have many values; for example, x could be 2 or x could be 0. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

From statement one, we know that x is either 2 or -1. From statement two, x + 3 is a prime number. If x = 2, x + 3 = 5 is a prime number. If x = -1, x + 3 = 2 is also a prime number. Thus, we still can’t determine the value of x. The two statements together are still not sufficient to answer the question.

Answer: E

(1) insufic
x^2 - 1 = x + 1, x^2 - x - 2 = 0,
(x-2)(x+1)=0, x={2,-1}

(2) insufic
x=2: x+3=2+3=5=prime
x=-1: x+3=-1+3=2=prime

(1/2) insufic
x=2: x+3=2+3=5=prime
x=-1: x+3=-1+3=2=prime

Ans (E)

Hi Bunuel please help me understand why this is not possible.

St 1.. x^2-1 = x+1
Or,(x+1)(x-1) = (x-1)
Or,(x+1) = (x-1)/(x-1)
Or, x+1 = 1
So x = 0.

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First of all, it should be (x + 1)(x - 1) = x + 1, not (x + 1)(x - 1) = x - 1.

Next, you cannot reduce (x + 1)(x - 1) = x + 1 by (x + 1) because (x + 1) can be 0 and we cannot divide by 0. By doing so you loose a root (a possible solution), namely x = -1.

What yo should do, is to re-arrange to get (x + 1)(x - 1) - (x + 1) = 0, then factor out x + 1 to get (x + 1)(x - 2) = 0. So, x = -1 and x = 2.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.

St. 1:
x^2 - 1 = x + 1
=> x^2 - x - 2 = 0
=> x^2 - 2x + x - 2 = 0
=> x(x-2) + 1(x-2) = 0
=> (x+1)(x-2) = 0
=> x = 2, x = -1
=> St. 1 is insufficient.

St. 2:
x + 3 -> prime
x = 2, 4, 8... and many more values (both positive and negative)
=> St. 2 is insufficient.

St. 1 & St. 2:
if x = -1, x+3 = 2 -> prime
if x = 2, x+3 = 5 -> prime
St. 1 & St. 2 are together insufficient.

Answer: E

What is the value of \(x\)?

(1) \(x^2-1=x+1\)

Rearrange to obtain \((x-1)(x+1)-(x+1)=0\);

Factor out \(x+1\) to obtain \((x+1)(x-2) = 0\). Either \(x = -1\) or \(x = 2\). Not sufficient.

(2) \(x + 3\) is a prime number. Clearly insufficient by itself.

(1) + (2) Both values from (1) satisfy statement (2) as well, so \(x\) can still be either -1 or 2. Not sufficient.


Answer: E