Bunuel wrote:
Competition Mode Question
How many integer values of x satisfy \(x(x + 2)(x + 4)(x + 6) < 200\)?
A. 6
B. 7
C. 8
D. 9
E. 10
If
x is 0, -2, -4, or -6, we see that the value of the expression on the left hand side of the expression will be 0 and thus less than 200.
If
x is -1, then one factor (the first one) is negative while the other three factors are positive. Therefore, the value of the expression is negative and less than 200. Similarly, if
x is -5, then one factor (the last one) is positive while the other three factors are negative. Therefore, the value of the expression is negative and less than 200.
If
x is -3, then the first two factors are negative and the last two are positive. Therefore, the value of the expression is positive. It might or might not be less than 200, so we have to check:
-3(-1)(1)(3) = 9 < 200 → Yes!
If x is greater than 0, then all the factors are positive. Therefore,the value of the expression is positive. It might or might not be less than 200, again we have to check:
x = 1: 1(3)(5)(7) = 105 < 200 → Yes!
x = 2: 2(4)(6)(8) = 8(48) < 200 → No!
We see that we don’t need to check any integer values greater than 2 since the value of the expression is already greater than 200 when x = 2. Last but not least, if x is less than -6, then all the factors are negative. Therefore,the value of the expression is positive. It might or might not be less than 200, so we have to check:
x = -7: -7(-5)(-3)(-1) = 105 < 200 → Yes!
x = -8: -8(-6)(-4)(-2) = 48(8) < 200 → No!
Again, we see that we don’t need to check any integer values greater than -8 since the value of the expression is already greater than 200 when x = -8. Therefore, there are 9 integer values (-7 to 1, inclusive) that satisfy the given inequality.
Answer: D