Answer
$(-3,3)$
Work Step by Step
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$x^4-7x^2-18\lt 0,$ let's let $x^2=k$
$k^2-7k-18\lt 0$,
$k^2+2k-9k-18\lt0$,
$k(k+2)-9(k+2)\lt0,$
$(k-9)(k+2)\lt0,$ substituing back in $x^2=k$.
$(x^2+2)(x^2-9)\lt0,$
$(x^2+2)(x-3)(x+3)\lt0,$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x^2+2)(x-3)(x+3) =0$
$x=-3$ or $x=3$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& &(a^2+2)(a-3)(a+3) & \\
(-\infty, -3) & -5 & (+)(-)(-)=(+) & F\\
(-3, 3) & 0 & (+)(-)(+)=(-) & T\\
(3,\infty) & 5 & (+)(+)(+)=(+) & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $(-3,3)$
