We are given the inequality:

\[
(x - 1)(3x + 2) \leq 2x - 2x + y
\]

This simplifies to:

\[
(x - 1)(3x + 2) \leq y
\]

Let's expand the left-hand side of the inequality:

\[
(x - 1)(3x + 2) = x(3x + 2) - 1(3x + 2) = 3x^2 + 2x - 3x - 2 = 3x^2 - x - 2
\]

Now the inequality becomes:

\[
3x^2 - x - 2 \leq y
\]

This is the simplified form of the inequality. 

### Interpretation:
This inequality represents a quadratic curve \( 3x^2 - x - 2 \) that lies below or equal to \( y \). The graph of this curve will determine the boundary, and \( y \) can take any value greater than or equal to this curve.

Would you like to explore graphing or solving this inequality further?

Here are 5 relative questions to consider:
1. What are the roots of the quadratic equation \( 3x^2 - x - 2 = y \)?
2. How does the value of \( y \) affect the solution set for \( x \)?
3. What happens when \( y \) is a constant (say, \( y = 0 \))?
4. How would we express the inequality solution in interval notation?
5. Can this inequality be solved using factoring or the quadratic formula?

**Tip:** When expanding polynomials, always ensure that all terms are combined properly to avoid errors in simplification.

Explore how to solve the quadratic inequality (x - 1)(3x + 2) ≤ y with a detailed step-by-step solution. This problem involves expanding and simplifying quadratic expressions, and understanding how quadratic curves relate to inequality solutions. Suitable for students in Grades 9-11.

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