Thread: Solving Inequality in Interval Notation

1. Solving Inequality in Interval Notation

Solve the following inequality. Write the answer in interval notation. Note If the answer includes more than one interval write the intervals separated by the "union" symbol, U. If needed, enter (infinity sign) as infinity and -(infinity sign) as -infinity.

x^3 - 16x </=0

I realize this might be easy for you guys but it isn't obvious to me and an explanation would be greatly appreciated!

2. It's $x(x+4)(x-4)\le 0.$
Study each factor for $(-\infty,-4),$ $(-4,0),$ $(0,4)$ and on $(4,\infty).$
For example $x$ is negative for the first interval, $x+4$ too and $x-4$ too, so we have three factors whose signs are negative, respectively, and hence its product is negative, so $x(x+4)(x-4)\le 0$ for the first interval. (Including the $-4,$ which is obviously a solution too.)