Problem 6

Explanation

Given $1 < x < y < 2$. We want to find the range of values for $z = x - y$.

Since $x < y$, the difference $x - y$ must be negative ($z < 0$).

Let's analyze the limits of $z$:

1. Maximum value of $z$

   The value of $z$ will be maximum (closest to $0$) when the difference between $x$ and $y$ is as small as possible.

   Since $x$ and $y$ are in the interval $(1, 2)$ and $x < y$, their difference can be arbitrarily close to $0$ but never reaches $0$ (because $x \neq y$).

   Thus, the upper bound of $z$ is $0$.

2. Minimum value of $z$

   The value of $z$ will be minimum (most negative) when $x$ is as small as possible and $y$ is as large as possible.

   • $x$ approaches $1$ (lower bound).
   • $y$ approaches $2$ (upper bound).

   Therefore:

   $$z \approx 1 - 2 = -1$$

   Thus, the lower bound of $z$ is $-1$.

Conclusion

The value of $z$ is between $-1$ and $0$.

In mathematical notation: $-1 < z < 0$.

Therefore, $z$ is between the values $-1$ and $0$.