Answer
$(-\infty,-7)\cup(-1,1)$
Work Step by Step
Step 1. Factor the inequality as
$x^2(x+7)-(x+7)\lt0$
or
$(x+7)(x+1)(x-1)\lt0$
Then the boundary points are $x=-7,-1,1$
Step 2. Using test points to examine signs across the boundary points, we have
$...(-)...(-7)...(+)...(-1)...(-)...(1)...(+)...$
Thus, the solutions are:
$x\lt-7$ plus $-1\lt x\lt1$
Step 3. We can express the solutions on a real number line as shown in the figure.
Step 4. We can express the solutions in interval notation as $(-\infty,-7)\cup(-1,1)$
