What is the length of the segment of the number line consisting of the points that satisfy $(x-4)^2 \le 9$?
I need to find the values of x that make the inequality $(x-4)^2 \le 9$ true. One way to do this is to take the square root of both sides and use the property that $\sqrt{a^2} = |a|$. This gives me $|x-4| \le 3$. This means that the distance between x and 4 is at most 3 units. To find the smallest and largest possible values of x, I can subtract and add 3 to 4. This gives me $x \in [1,7]$. The length of this interval is the difference between the endpoints, which is $7-1 = 6$. 