### Warm-up:

### The Problem:

Given that the mean of this set of numbers is 10, what are the possible ranges?

$$\{2x-7,x,{(x-1)}^{2},{x}^{2},5x\}$$-/|\-

### The Solution:

Using the definition of mean and the set of numbers given we get:

$$\frac{(2x-7+x+{(x-1)}^{2}+{x}^{2}+5x)}{5}=10$$Simplifying ...

$$2x-7+x+{(x-1)}^{2}+{x}^{2}+5x=50$$ $$8x-7+({x}^{2}-2x+1)+{x}^{2}=50$$ $$2{x}^{2}-6x-56=0$$This is a quadratics. Simplifying and solving ...

$$x=\{-4,7\}$$So, this gives us two posibilities for the range.

For $x=-4$ gives this set of values $\{-15,-4,25,16,-20\}$ resulting in a range of $25--20=45$

For $x=7$ gives this set of values $\{7,7,36,49,35\}$ resulting in a range of $49-7=42$

-oOo-