BrentGMATPrepNow wrote:
If \(x\) is a positive integer, is \(x > 4\)?
(1) \(x^2 + 32 = 12x\)
(2) \(x^x = \sqrt{x^{16}}\)
Given: \(x\) is a positive integer Target question: Is \(x > 4\)? Statement 1: \(x^2 + 32 = 12x\) Since we are given a quadratic equation let's first set it equal to zero: \(x^2 - 12x + 32 = 0\)
Factor to get: \((x - 4)(x - 8) = 0\), which means EITHER \(x = 4\) OR \(x =8\).
Case a: If \(x = 4\), the answer to the target question is
NO, x is not greater than 4Case b: If \(x = 8\), the answer to the target question is
YES, x is greater than 4Since we can’t answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(x^x = \sqrt{x^{16}}\)Rewrite the equation as follows: \(x^x = (x^{16})^{\frac{1}{2}}\)
- Taking the square root of a value is equivalent to raising that value to the power of \(\frac{1}{2}\)Simplify the right side to get: \(x^x = x^8\)
Note: I created this question to remind students that, when it comes to equations with variables in the exponents, there are three important provisos we must consider before we can conclude that two exponents are equal.
That is, if \(x^a = x^b\), then we can conclude that \(a = b\) AS LONG AS \(x \neq 0\), \(x \neq 1\), and \(x \neq -1\).
For example, if we know that \(0^x = 0^3\), we can't then conclude that \(x = 3\), since there are infinitely many values of \(x\) that satisfy the equation. So, if \(x\) does NOT equal -1, 0 or 1, then we can conclude that \(x = 8\)
Now let's test the forbidden numbers (i.e., -1, 0 and 1).
Since we're told
x is positive, we need only test whether \(x = 1\) satisfies the given equation.
Plug \(x = 1\) into the equation \(x^x = x^8\) to get: \(1^1 = 1^8\)....IT WORKS!!
So, \(x = 1\) is another solution to the equation.
Case a: If \(x = 8\), the answer to the target question is
YES, x is greater than 4Case b: If \(x = 1\), the answer to the target question is
NO, x is not greater than 4Since we can’t answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1 tells us that either \(x = 4\) or \(x =8\)
Statement 2 tells us that either \(x = 1\) or \(x =8\)
Since \(x =8\) is the only x-value the two statements have in common, we can be certain that \(x =8\), which means the answer to the target question is
YES, x is greater than 4Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C