Answer
The inequality is valid for values less than 4 and more than 6 (including them) i.e. $(-\infty,4)\cap (6,\infty)$
Work Step by Step
First, we are going to factor to find the x-intercepts:
$x^2-10x+24=0$
$(x-4)(x-6)=0$
$x_1=4$
$x_2=6$
These are the critical points. We are going to take three values: one less than 4, one between 4 and 6, and one more than 6 to test in the original equation and check if the inequality is true or not:
First test with a value less than 4:
$1^2-10(1)+24\geq0$
$1-10+24\geq0$
$15\geq0 \rightarrow \text{ TRUE}$
Second test with a value between 4 and 6:
$5^2-10(5)+24\geq0$
$25-50+24\geq0$
$-1\geq0 \rightarrow \text{ FALSE}$
Third test with a value more than 6:
$10^2-10(10)+24\geq0$
$100-100+24\geq0$
$24\geq0 \rightarrow \text{ TRUE}$
These tests show that the inequality $x^2-10x+24\geq0$ is valid for values less than 4 and more than 6 (including them) i.e. $(-\infty,4)\cap (6,\infty)$
