Answer
$(-\infty,4]\cup[6,\infty)$
Work Step by Step
We are given the inequality:
$x^2-10x+24\geq 0$
Consider the function $f(x)=x^2-10x+24$.
Determine the $x$-intercepts of $f$ by setting $y=0$ then solving for $x$:
$x^2-10x+24=0$
Factor the trinomial by looking for factors of $24$ whose sum is $-10$ to obtain:
$(x-6)(x-4)=0$
Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
$x-6=0$ or $x-4=0$
$x=6$ or $x=4$
Determine the vertex using the formula $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ to obtain:
$-\dfrac{b}{2a}=-\dfrac{-10}{2(1)}=5$
$f\left(-\frac{2}{2a}\right)=f(5)=5^2-10(5)+24=25-50+24=-1$
Therefore the vertex is at $(5,-1)$.
Graph the function. (refer to the grah below).
The solution of the inequality consists of the values of $x$ for which the graph of $f(x)$ is above or on the $x$-axis:
Notice the the graph of $f(x)$ is above or on the $x$ axis in the following intervals:
$(-\infty,4]\cup[6,\infty)$
