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.
$x^4-7x^2-18 \lt 0$,
$x^4+2x^2-9x^2-18 \lt 0$,
$x^2(x^2+2)-9(x^2+2) \lt 0$,
$(x^2-9)(x^2+2) \lt 0$,
$(x-3)(x+3)(x^2+2) \lt 0$
$f(x)=(x-3)(x+3)(x^2+2)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-3)(x+3)(x^2+2)=0$
$x=3$ 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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& &(a-3)(a+3)(a^2+2)& \\
(-\infty, -3) & -4 & (-)(-)(+) & F\\
(-3,3) & 0 & (-)(+)(+) & T\\
(3,\infty) & 4 & (+)(+)(+) & F
\end{array}$
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: $(-3,3)$
