# Chapter 8 - Section 8.5 - Polynomial and Rational Inequalities - Exercise Set - Page 649: 5

$$(-∞, 1) ∪ (4,+∞)$$ 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: $$(-∞, 1) ∪ (4,+∞)$$ 