Answer
$$[2, 4]$$
Work Step by Step
Let $x^2 -6x +8 \leq 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 -6x +8=0$
Factor: $x^2 -6x +8$ = $(x-2)(x-4)$
$x-2=0$ or $x-4=0$
$x = 2$ 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:
$(-∞, 2)(2,4)(4, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, 2)$
Test Value = $0$
$f=(x-2)(x-4)$
$f=(0-2)(0-4)$
$f=8$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, 2)$.
$(2, 4)$
Test Value = $3$
$f=(x-2)(x-4)$
$f=(3-2)(3-4)$
$f=-1$
Conclusion: $f (x) < 0$ for all $x$ in $(2,4)$.
$(4,+∞)$
Test Value = $5$
$f=(x-2)(x-4)$
$f=(5-2)(5-4)$
$f=3$
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 $f (x) \leq 0$, where $f (x) = x^2 -6x +8$. Based on the solution above, $f(x)\leq0$ for all $x$ in $(2, 4)$. However, because the inequality involves $\leq$ (less than or equal to), we must also include the solutions of $x^2 -6x +8 = 0$, namely $2$ and $4$, in the solution set.
Thus, the solution set of the given inequality is:
$$[2, 4]$$
