Answer
$$(-∞, -3) ∪ (5,+∞)$$
Work Step by Step
Let $(x + 3)(x - 5) > 0$ be a function $f$.
Find the $x$-intercepts by solving $(x + 3)(x - 5) > 0=0$
$x+3=0$ or $x-5=0$
$x = -3$ or $x = 5$
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:
$(-∞, -3)(-3,5)(5, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -3)$
Test Value = $-4$
$f=(x + 3)(x - 5)$
$f=(-4+3)(-4-5)$
$f=9$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -3)$.
$(-3,5)$
Test Value = $0$
$f=(x + 3)(x - 5)$
$f=(0+3)(0-5)$
$f=-15$
Conclusion: $f (x) < 0$ for all $x$ in $(-3,5)$.
$(5,+∞)$
Test Value = 6
$f=(x + 3)(x - 5)$
$f=(6+3)(6-5)$
$f=9$
Conclusion: $f (x) > 0$ for all $x$ in $(5,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given
inequality. We are interested in solving $f (x) > 0$, where $f (x) = (x+3)(x-5)$. Based on the solution above, $f(x)>0$ for all $x$ in $(-∞, -3)$ or $(5,+∞)$.
Thus, the solution set of the given inequality is:
$$(-∞, -3) ∪ (5,+∞)$$
