Answer
$$(-∞, -3) ∪ (2,+∞)$$
Work Step by Step
Let $x^2 +x -6 \gt 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 +x -6=0$
Factor: $x^2 +x -6$ = $(x-2)(x+3)$
$x-2=0$ or $x+3=0$
$x = 2$ or $x = -3$
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:
$(-∞, -3)(-3,2)(2, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -3)$
Test Value = $-4$
$f=(x-2)(x+3)$
$f=(-4-2)(-4+3)$
$f=6$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -3)$.
$(-3,2)$
Test Value = $0$
$f=(x-2)(x+3)$
$f=(0-2)(0+3)$
$f=-6$
Conclusion: $f (x) < 0$ for all $x$ in $(-3,2)$.
$(2,+∞)$
Test Value = $3$
$f=(x-2)(x+3)$
$f=(3-2)(3+3)$
$f=6$
Conclusion: $f (x) > 0$ for all $x$ in $(2,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) > 0$, where $f(x) = x^2 +x - 6$. Based on the solution above, $f(x)>0$ for all $x$ in $(-∞, -3)$ or $(2,+∞)$.
Thus, the solution set of the given inequality is:
$$(-∞, -3) ∪ (2,+∞)$$
