# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-10/8

Exercises: Try Out - Set 10: Real exam simulation to sharpen your skills and build confidence. - Quadratic Inequality

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## Exercise 8

### Question

export const metadata = {
  title: "Quadratic Inequality",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given:

1.  <InlineMath math="x > 2" />
2.  <InlineMath math="-3 < x < 2" />
3.  <InlineMath math="x < -3" />
4.  <InlineMath math="x > -3" />

If <InlineMath math="x^2 + x - 6 > 0" />, then the inequality is satisfied by...


### Choices

- [x] $$1$$ and $$3$$
- [ ] $$2$$ and $$4$$
- [ ] $$1$$ and $$4$$
- [ ] $$3$$ and $$4$$
- [ ] $$1$$

### Answer & Explanation

import { Graph } from "../graph";

export const metadata = {
  title: "Quadratic Inequality Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

We will solve the inequality:

<MathContainer>
  <BlockMath math="x^2 + x - 6 > 0" />
</MathContainer>

Factor the quadratic equation:

<MathContainer>
  <BlockMath math="(x - 2)(x + 3) > 0" />
</MathContainer>

The zeros (roots) of the equation are:

<MathContainer>
  <BlockMath math="x = 2 \quad \text{or} \quad x = -3" />
</MathContainer>

We test points on the number line:

<Graph
  title="Number Line"
  description="Visualization of the inequality solution region."
/>

-   For <InlineMath math="x < -3" /> (e.g., <InlineMath math="-4" />): <InlineMath math="(-4-2)(-4+3) = (-6)(-1) = 6" /> (Positive)
-   For <InlineMath math="-3 < x < 2" /> (e.g., <InlineMath math="0" />): <InlineMath math="(0-2)(0+3) = (-2)(3) = -6" /> (Negative)
-   For <InlineMath math="x > 2" /> (e.g., <InlineMath math="3" />): <InlineMath math="(3-2)(3+3) = (1)(6) = 6" /> (Positive)

Since the inequality sign is <InlineMath math="> 0" /> (positive), the solution region is:

<MathContainer>
  <BlockMath math="x < -3 \quad \text{or} \quad x > 2" />
</MathContainer>

Based on the given statements:
1.  <InlineMath math="x > 2" /> (Matches)
2.  <InlineMath math="-3 < x < 2" /> (Does not match)
3.  <InlineMath math="x < -3" /> (Matches)
4.  <InlineMath math="x > -3" /> (Does not match)

Thus, the inequality is satisfied by statements <InlineMath math="1" /> and <InlineMath math="3" />.


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