M03-01

Official Solution:


Is \(x > 3\)?

(1) \((x - 3)(x - 2)(x - 1) > 0\)

Here is how to solve the above inequality easily. The "roots", in ascending order, are 1, 2, and 3, which gives us 4 ranges:

Next, test an extreme value for \(x\): if \(x\) is some large enough number, say 10, then all three multiples will be positive, giving a positive result for the whole expression. So when \(x > 3\), the expression is positive. Now the trick: as in the 4th range, the expression is positive, then in the 3rd it'll be negative, in the 2nd it'll be positive, and finally in the 1st, it'll be negative: \(\text{(- + - +)}\). So, the ranges when the expression is positive are: \(1 < x < 2\) and \(3 < x\).

Therefore, \(x\) may or may not be greater than 3. Not sufficient.

P.S. You can apply this technique to any inequality of the form \((ax - b)(cx - d)(ex - f)... > 0\) or \( < 0\). If one of the factors is in the form \((b - ax)\) instead of \((ax - b)\), simply rewrite it as \(-(ax - b)\) and adjust the inequality sign accordingly. For example, consider the inequality \((4 - x)(2 + x) > 0\). Rewrite it as \(-(x - 4)(2 + x) > 0\). Multiply by -1 and flip the sign: \((x - 4)(2 + x) < 0\). The "roots", in ascending order, are -2 and 4, giving us three ranges: \(x < -2\), \(-2 < x < 4\), and \(x > 4\). If \(x\) is some large enough number, say 100, both factors will be positive, yielding a positive result for the entire expression. Thus, when \(x > 4\), the expression is positive. Now, apply the alternating sign trick: as the expression is positive in the 3rd range, it will be negative in the 2nd range, and positive in the 1st range: \(\text{(+ - +)}\). Therefore, the ranges where the expression is negative are: \(-2 < x < 4\).

(2) \(x > 1\)

\(x\) may or may not be greater than 3. Not sufficient.

(1)+(2) Combining both statements, the second statement does not narrow down the range of \(x\) obtained from the first statement. We still have \(1 < x < 2\) and \(3 < x\). Consequently, we cannot determine if \(x > 3\) based on the available information. Not sufficient.


Answer: E