Answer
$$(-∞, 0] ∪ [4,+∞)$$
Work Step by Step
Let $x^2 - 4x \geq 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 - 4x=0$
Factor: $x^2 - 4x = x(x-4)$
$x=0$ or $x-4=0$
$x = 0$ or $x = 4$
These $x$-intercepts serve as boundary points that separate the number line into intervals.
The boundary points of this equation therefore divide the number line into three intervals:
$(-∞, 0)(0,4)(4, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, 0)$
Test Value = $-1$
$f=x(x-4)$
$f=(-1)(-1-4)$
$f=5$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, 0)$.
$(0, 4)$
Test Value = $1$
$f=x(x-4)$
$f=(1)(1-4)$
$f=-3$
Conclusion: $f (x) < 0$ for all $x$ in $(0, 4)$.
$(4,+∞)$
Test Value = $5$
$f=x(x-4)$
$f=(5)(5-4)$
$f=5$
Conclusion: $f (x) > 0$ for all $x$ in $(4,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) \geq 0$, where $f (x) = x^2-4x$. Based on the solution above, $f(x)\gt0$ for all $x$ in $(-∞,0)$ or $(4, +∞)$. However, because the inequality involves $\geq$ (greater than or equal to), we must also include the solutions of $x^2-4x = 0$, namely $0$ and $4$, in the solution set.
Thus, the solution set of the given inequality is:
$$(-∞, 0] ∪ [4,+∞)$$
