53. How many integers in the inequality vert x^2-xvert gt 2x Does it not apply? E) 6 A) 2 B) 3 C) 4 (D)

To solve the inequality , we need to consider two cases based on the absolute value expression.**Case 1:** This simplifies to: Factor the quadratic: The critical points are and . We test intervals around these points:- For - For , choose : \(4(4 - 3) = 4 > 0\)Thus, the solution for this case is \(x \in (-\infty, 0) \cup (3, \infty)\).**Case 2:** \(-(x^2 - x) > 2x\) or equivalently - For - For , choose : \(1(1 + 1) = 2 > 0\)Thus, the solution for this case is \(x \in (-1, 0)\).Combining both cases, the solution to the inequality is: Now, we find the integers that do not satisfy this inequality. These integers are in the intervals where the inequality does not hold, which are: The integers in these intervals are: .Therefore, there are 3 integers where the inequality does not apply. The answer is:B) 3