Solve the inequality.` (x-6)^2(x+1)>0`Express solution in interval notation.
Well, to solve this, you need to be a bit clever. We know that our solution set will not allow 0 to be a solution. Therefore, we know the following:
`x-6 != 0`
`x+1 != 0`
Otherwise the left side would be zero! Now, let's consider some other facts about this. Notice the squared term. Can a squared term ever be less than 0? Let's graph `(x-6)^2` and see:
Well, it looks like it hits 0 at 6, but the graph never goes below zero. Therefore, the `(x-6)^2` term does not matter for our inequality! We can simply divide it out:
`x+1 < 0`
Now, we're left with a pretty simple inequality! Let's solve by subtracting 1:
`x < -1`
Well, looking at our constraints (that x not be equal to 1 or 6 based on our first equations above), this solution set fits our constraints. So, let's graph the full equation `y = (x-6)^2(x+1)` and the part of the graph that follows from our solution set (`x<-1`) to see if we're correct. The full graph is in black. Our solution is in red.
Looks like we're correct! The red line is the only part of the graph that is negative, as the question asks for.
The solution can also be expressed the following way: