How many non-zero integers satisfy the inequality x2 - 5|x| - 6 ≤ 0?

To solve the inequality x^2 - 5|x| - 6 ≤ 0, we need to consider two cases because of the absolute value |x|.

Case 1: x ≥ 0 The inequality becomes x^2 - 5x - 6 ≤ 0. We can factor this into (x-6)(x+1) ≤ 0. The solutions to this inequality are -1 ≤ x ≤ 6. But since we're considering the case where x ≥ 0, the solutions are 0 ≤ x ≤ 6. Since we're looking for non-zero integers, the solutions in this case are x = 1, 2, 3, 4, 5, 6.

Case 2: x < 0 The inequality becomes x^2 + 5x - 6 ≤ 0. We can factor this into (x+6)(x-1) ≤ 0. The solutions to this inequality are -6 ≤ x ≤ 1. But since we're considering the case where x < 0, the solutions are -6 ≤ x < 0. Since we're looking for non-zero integers, the solutions in this case are x = -1, -2, -3, -4, -5, -6.

Combining both cases, the non-zero integers that satisfy the inequality are x = -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6. So, there are 12 non-zero integers that satisfy the inequality.