Question

How do you solve `x^ { 2} - 2x - 24\geq x - 6`?

Answer 1

Solution set:
(-inf., -3] and [6, +inf.)

Explanation:

Bring the inequality to the standard form of a quadratic inequality:
@f(x) = x^2 - 3x - 18 >= 0#.
First, solve f(x) = 0
Find 2 real roots knowing the sum (-b = 3) and the product (c = - 18). They are -3 and 6.
The parabola graph of f(x) opens upward because a > 0. Inside the interval (-3, 6), the graph is below the x-axis, meaning f(x) < 0. Therefor, f(x) > 0 outside this interval. The 2 end points are included in the solution set.
Answers by 2 half closed intervals: (-inf., -3] and [6, +inf.)
Answer by the number line:

++++++++++++ -3 ------------ 0 ---------------------- 6 ++++++++++++