# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-9/6

Exercises: Try Out - Set 9: Real exam simulation to sharpen your skills and build confidence. - Problem 6

---

## Exercise 6

### Question

export const metadata = {
  title: "Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

If <InlineMath math="1 < x < y < 2" /> and <InlineMath math="z = x - y" />, then <InlineMath math="z" /> is between the values...


### Choices

- [ ] $$0$$ and $$2$$
- [ ] $$1$$ and $$2$$
- [x] $$-1$$ and $$0$$
- [ ] $$-2$$ and $$2$$
- [ ] $$-2$$ and $$1$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

Given <InlineMath math="1 < x < y < 2" />. We want to find the range of values for <InlineMath math="z = x - y" />.

Since <InlineMath math="x < y" />, the difference <InlineMath math="x - y" /> must be negative (<InlineMath math="z < 0" />).

Let's analyze the limits of <InlineMath math="z" />:

1.  **Maximum value of <InlineMath math="z" />**

    The value of <InlineMath math="z" /> will be maximum (closest to <InlineMath math="0" />) when the difference between <InlineMath math="x" /> and <InlineMath math="y" /> is as small as possible.

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

    Thus, the upper bound of <InlineMath math="z" /> is <InlineMath math="0" />.

2.  **Minimum value of <InlineMath math="z" />**

    The value of <InlineMath math="z" /> will be minimum (most negative) when <InlineMath math="x" /> is as small as possible and <InlineMath math="y" /> is as large as possible.

    -   <InlineMath math="x" /> approaches <InlineMath math="1" /> (lower bound).
    -   <InlineMath math="y" /> approaches <InlineMath math="2" /> (upper bound).

    Therefore:

    <MathContainer>
      <BlockMath math="z \approx 1 - 2 = -1" />
    </MathContainer>

    Thus, the lower bound of <InlineMath math="z" /> is <InlineMath math="-1" />.

**Conclusion**

The value of <InlineMath math="z" /> is between <InlineMath math="-1" /> and <InlineMath math="0" />.

In mathematical notation: <InlineMath math="-1 < z < 0" />.

Therefore, <InlineMath math="z" /> is between the values <InlineMath math="-1" /> and <InlineMath math="0" />.


---
