# Chapter 8 - Section 8.5 - Polynomial and Rational Inequalities - Exercise Set - Page 649: 7

$$(-∞, -4) ∪ (-1,+∞)$$ 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: $(-∞, -4)(-4,-1)(-1, +∞)$ Choose one test value within each interval and evaluate $f$ at that number. $(-∞, -4)$ Test Value = $-5$ $f=(x+1)(x+4)$ $f=(5+1)(5+4)$ $f=45$ Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -4)$. $(-4,-1)$ Test Value = $-3$ $f=(x+1)(x+4)$ $f=(-3+1)(-3+4)$ $f=-2$ Conclusion: $f (x) < 0$ for all $x$ in $(-4,-1)$. $(-1,+∞)$ Test Value = $0$ $f= (x+1)(x+4)$ $f=(0+1)(0+4)$ $f=4$ Conclusion: $f (x) > 0$ for all $x$ in $(-1,+∞)$. 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 $(-∞,-4)$ or $(-1, +∞)$. Thus, the solution set of the given inequality is: $$(-∞, -4) ∪ (-1,+∞)$$ 