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Since absolute values are always non-negative, we know that the numerator will always be non-negative. Therefore, for the expression to be negative, the denominator must be negative and the numerator must be strictly positive. Factoring the denominator gives us
x 2 − 3 6 x = x ( x − 3 6 )
The zeroes of this expression are 0 and 36.
If x > 3 6 , both factors x and x − 3 6 are positive, and their product is positive.
If x < 0 , both factors are negative, and their product is positive.
If 0 < x < 3 6 , the factor x is positive and the factor x − 3 6 is negative, and their product is negative.
So we've narrowed down the possible solutions to the integers from 1 to 35 inclusive. Now we just need to determine if any of these integers make the numerator zero. Solving the equation x 2 − 8 1 = 0 for x , we find that the numerator is zero when x = ± 9 . Therefore, the solutions are all of the integers from 1 to 35 inclusive except 9, so there are 34 integer values of x that satisfy the inequality.