Answer:
To write a possible quadratic inequality, we first need to determine which quadratic function has roots at x = -0.5 and x = 1.5.
We know that the solution to the quadratic inequality is x < -0.5 or x > 1.5, which means that the parabola associated with our quadratic function should not intersect the x-axis between x = -0.5 and x = 1.5.
One quadratic function with roots at x = -0.5 and x = 1.5 is:
(x + 0.5)(x - 1.5) ≤ 0
To see why this is the case, we can use a graphing calculator to graph the quadratic function y = (x + 0.5)(x - 1.5) and observe that the function is negative (i.e., has a y-value less than zero) when x is less than -0.5 or greater than 1.5.
Alternatively, we can use the fact that the product of two factors is less than or equal to zero if and only if one factor is positive and the other is negative.
Since (x + 0.5) is positive for x > -0.5 and negative for x < -0.5, and (x - 1.5) is positive for x > 1.5 and negative for x < 1.5, the inequality (x + 0.5)(x - 1.5) ≤ 0 is true for x < -0.5 or x > 1.5, but false for -0.5 < x < 1.5.
Therefore, a possible quadratic inequality that satisfies the given set of values for x is:
(x + 0.5)(x - 1.5) ≤ 0