#### Answer

$$(-∞, 1) ∪ (4,+∞)$$

#### Work Step by Step

Let $x^2 - 5x + 4 \gt 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 - 5x + 4=0$
Factor: $x^2 - 5x + 4$ = $(x-1)(x-4)$
$x-1=0$ or $x-4=0$
$x = 1$ 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:
$$(-∞, 1)(1,4)(4, +∞)$$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, 1)$
Test Value = $0$
$f=(x-1)(x-4)$
$f=(0-1)(0-4)$
$f=4$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, 1)$.
$(1,4)$
Test Value = $2$
$f=(x-1)(x-4)$
$f=(2-1)(2-4)$
$f=-2$
Conclusion: $f (x) < 0$ for all $x$ in $(1,4)$.
$(4,+∞)$
Test Value = $5$
$f=(x-1)(x-4)$
$f=(5-1)(5-4)$
$f=4$
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 $f (x) > 0$, where $f(x) = x^2 - 5x + 4$. Based on the solution above, $f(x)>0$ for all $x$ in $(-∞, 1)$ or $(4,+∞)$.
Thus, the solution set of the given inequality is:
$$(-∞, 1) ∪ (4,+∞)$$