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Statement 1: z² < y No information about x, so statement 1 is NOT SUFFICIENT
Statement 2: x < z No information about y, so statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined NOTE: the error that many students will make is to assume that z < z², but this is not necessarily the case. For example, if z = 1/2, then z² = 1/4, in which case z² < z On the other hand, if z = 3, then z² = 3, in which case z < z² This feature will create conflicting answers to the target question. Here's what I mean: There are several values of x, y and z that satisfy BOTH statements. Here are two: Case a: x = 0.3, y = 0.4, and z = 0.5 (which also means z² = 0.25). Notice that z² < y and x < z, so both conditions are met. In this case x < y Case b: x = 0.4, y = 0.3, and z = 0.5 (which also means z² = 0.25). Notice that z² < y and x < z, so both conditions are met. In this case x > y Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT
Since we do not have any information regarding x, statement one is not sufficient to answer the question.
Statement Two Alone:
x < z
Since we do not have any information regarding y, statement two is not sufficient to answer the question.
Statements One and Two Together:
Using the information in statements one and two, we still cannot answer the question. For instance, if y = 10, z = 2, and x = 1, then x is less than y. However, if z = 1/2, x = 1/3, and y = 1/3, then x IS NOT less than y.
Statement 1 : We don't know the value of x therefore not sufficient. Statement 2 : We don't know the value of y therefore not sufficient.
Together Statement 1 and 2, ( Statement 1 says z^2 < y ... let z be 0.5 then z^2 = 0.25 ... and y is greater than z^2 let y be 0.26 onwards.... ( SO Z= 0.5 AND Y = 0.26 ) Statement 2 says z is greater than x which means x is less than 0.5 which means X could be 0.4
In this case X > y ( 0.4 > 0.26 ) therfore is Is x < y ? ans is not true.
nOW LETS TEST THIIS with another value .. But now if we assume the value of z = 2 then z^2 = 2^2 = 4 and y should be greater than 4 lets say 5 ... so Z= 2 and y = 5
Statement 2 says x < Z so lets say x = 1 in that case x< y or 1 < 5 true ...
Since we cannot answer the question with certainty ans is E
1) and 2) are insufficient alone. We can combine the statements by squaring 2) to get z^2.
However, depending on the sign of x and z we can have \(x^2 < z^2\) or \(x^2 > z^2\) so we must entertain both possibilities (at this point a good guess is E because we already have two cases).
In the first case, we have \(x^2 < z^2 < y\), thus \(x^2 < y\). This cannot simplify to \(x < y\) because we can have x = 0.8 and y = 0.7. Thus combining is insufficient. Knowing that \(x^2 < y\) cannot simplify to \(x < y\) can be valuable knowledge for future problems. Since we cannot prove \(x < y\) with the given information it is insufficient.
The second case occurs happens when x is negative and has a bigger magnitude than z. Hence x must be negative in this case. From (1) we know y is positive, so \(x < y\).
However, since the first case was already insufficient, the statements combined are insufficient, without needing to work out case 2.
Ans: E
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