solve quadratic inequality 
y < x² + 2x - 4

To solve the quadratic inequality \( y < x^2 + 2x - 4 \), we follow these steps:

### Step 1: Understand the inequality
The given inequality is \( y < x^2 + 2x - 4 \), which describes the region below the parabola defined by \( y = x^2 + 2x - 4 \).

### Step 2: Analyze the quadratic expression
The quadratic expression \( x^2 + 2x - 4 \) represents a parabola that opens upwards since the coefficient of \( x^2 \) is positive. The vertex of this parabola occurs at the point where the derivative \( 2x + 2 = 0 \), giving \( x = -1 \).

Substitute \( x = -1 \) into the equation to find the vertex:
\[
y = (-1)^2 + 2(-1) - 4 = 1 - 2 - 4 = -5
\]
So, the vertex of the parabola is at \( (-1, -5) \).

### Step 3: Graph the inequality
- The equation \( y = x^2 + 2x - 4 \) represents a parabola with its vertex at \( (-1, -5) \), and it opens upward.
- The inequality \( y < x^2 + 2x - 4 \) represents all the points **below** this parabola.

### Step 4: Interpret the solution
- In terms of \( x \), there is no specific range for \( x \) values where the inequality holds. The solution is a region in the \( xy \)-plane where the \( y \)-values are less than the \( y \)-values on the parabola.

### Final Answer:
The solution to the inequality \( y < x^2 + 2x - 4 \) is the set of all points \( (x, y) \) in the plane where the \( y \)-coordinate is less than the value of \( x^2 + 2x - 4 \) for a given \( x \)-coordinate. 

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Here are 5 related questions to expand on this topic:

1. How do you solve quadratic inequalities in one variable?
2. What is the significance of the vertex in a parabola?
3. How do we determine the direction of a parabola's opening?
4. How do you graph quadratic inequalities?
5. What is the difference between solving quadratic inequalities and equations?

**Tip:** Always remember that the solution to a quadratic inequality involves regions in the coordinate plane, not just specific points!

solve quadratic inequality y < x² + 2x - 4

Learn how to solve the quadratic inequality y < x² + 2x - 4 by understanding the graph of the parabola and interpreting the region below it. This solution includes key algebraic concepts and step-by-step instructions for students in Grades 9-11.

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