Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. You reached 1 for an answer, didn’t you? How does this number game work? (Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression). How did the number game use the skill of simplifying rational expressions? Create your own number game using the rules of algebra and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions.

Assume the number is x.

Square that number and then subtract one: x^2 - 1.

Divide by one less than your original number: (x^2 - 1)/(x-1) = x+1

Now subtract your original number: x+1-x = 1.

The number game you described can be rewritten using a variable as follows:

Pick any number (except for 1), let's call it "x". Now, follow these steps:
1. Square that number: x^2
2. Subtract one: x^2 - 1
3. Divide by one less than your original number, which is (x - 1): (x^2 - 1)/(x - 1)
4. Finally, subtract your original number, x: (x^2 - 1)/(x - 1) - x

You will find that no matter what value of x you choose (except for 1), the final expression simplifies to 1. This can be proven algebraically:

Let's simplify the expression (x^2 - 1)/(x - 1) - x step by step:
1. Factor the numerator (x^2 - 1) as a difference of squares: (x - 1)(x + 1).
2. Combine the fractions: [(x - 1)(x + 1)]/(x - 1) - x.
3. Simplify by canceling out the factor (x - 1): (x + 1) - x.
4. Combine like terms: x + 1 - x = 1.

So, no matter what value of x you choose (except for 1), the expression simplifies to 1.

This number game demonstrates the skill of simplifying rational expressions because we simplified the expression (x^2 - 1)/(x - 1) - x by factoring and canceling common factors, leading to the simplified form of 1.

Now, let me create a number game for you using the rules of algebra:

In this number game, pick any number x (x ≠ 0). Now, follow these steps:
1. Multiply x by 2: 2x
2. Add 5: 2x + 5
3. Square the result: (2x + 5)^2
4. Subtract 25: (2x + 5)^2 - 25
5. Divide by x: [(2x + 5)^2 - 25]/x

The challenge for your classmates is to simplify the expression [(2x + 5)^2 - 25]/x as much as possible and find any restrictions on the values of x that would make the expression undefined.

This number game indeed uses the skill of simplifying rational expressions because the expression [(2x + 5)^2 - 25]/x can be simplified by factoring, canceling common factors, and simplifying any remaining terms.