What is the set of real solutions for |𝑥|2−|𝑥|−2=0?

The given equation is |𝑥|² - |𝑥| - 2 = 0.

Step 1: We can rewrite this equation as a quadratic equation. Let's denote |𝑥| as y. So, the equation becomes y² - y - 2 = 0.

Step 2: Now, we can solve this quadratic equation. The solutions of the quadratic equation ax² + bx + c = 0 are given by the formula: x = [-b ± sqrt(b² - 4ac)] / 2a.

In our case, a = 1, b = -1, and c = -2.

Step 3: Substituting these values into the formula, we get: y = [1 ± sqrt((-1)² - 41(-2))] / 2*1 = [1 ± sqrt(1 + 8)] / 2 = [1 ± sqrt(9)] / 2 = [1 ± 3] / 2.

Step 4: So, the solutions for y are y1 = (1 + 3) / 2 = 2 and y2 = (1 - 3) / 2 = -1.

Step 5: However, since y = |x|, it must be non-negative. Therefore, y cannot be -1. So, the only solution for y is y = 2.

Step 6: Finally, we need to find the solutions for x. Since y = |x| = 2, the solutions for x are x1 = 2 and x2 = -2.

So, the set of real solutions for the given equation is {2, -2}.