How many real solutions are there for the equation x2 – 7|x| - 30 = 0?

To solve this problem, we need to consider two cases because of the absolute value in the equation.

Case 1: x ≥ 0 The equation becomes x^2 - 7x - 30 = 0. This is a quadratic equation and can be factored to (x - 10)(x + 3) = 0. The solutions are x = 10 and x = -3. However, since we assumed x ≥ 0, x = -3 is not a valid solution. So, there is 1 solution in this case.

Case 2: x < 0 The equation becomes x^2 + 7x - 30 = 0. This is also a quadratic equation and can be factored to (x - 5)(x + 6) = 0. The solutions are x = 5 and x = -6. However, since we assumed x < 0, x = 5 is not a valid solution. So, there is 1 solution in this case.

Therefore, the equation x^2 – 7|x| - 30 = 0 has 2 real solutions: x = 10 and x = -6.