Given the quadratic equation y = x\(^2\) - x - 12, we can find the range of values of x for which y \( \geq \) 0 by finding the roots of the equation and determining the sign of y for values of x between and outside the roots.

The quadratic equation can be factored as y = (x - 4)(x + 3). Setting y = 0, we find that the roots of the equation are x = 4 and x = -3.

To determine the sign of y for values of x between and outside the roots, we can use a sign chart. We construct a sign chart by placing the roots on a number line and testing the sign of y for values of x in each interval defined by the roots.

For x < -3, we can choose a test value such as x = -4. Substituting this value into the equation, we get y = (-4)\(^2\) - (-4) - 12 = 16 + 4 - 12 = 8, which is positive. Therefore, y is positive for all values of x < -3.

For -3 < x < 4, we can choose a test value such as x = 0. Substituting this value into the equation, we get y = 0\(^2\) - 0 - 12 = -12, which is negative. Therefore, y is negative for all values of x in the interval (-3, 4).

For x > 4, we can choose a test value such as x = 5. Substituting this value into the equation, we get y = 5\(^2\) - 5 - 12 = 25 - 5 - 12 = 8, which is positive. Therefore, y is positive for all values of x > 4.

Since we want to find the range of values of x for which y \( \geq \) 0, we need to include the intervals where y is positive as well as the values of x where y is equal to zero (the roots). Therefore, the solution to the inequality y \( \geq \) 0 is x \( \leq \) -3 or x \( \geq \) 4