https://file.aiquickdraw.com/mathbot/file/172657818208948l26yu8.jpg

Let's go through each step to find where the error occurred in the student's solution of the quadratic inequality.

### Step-by-step analysis:

1. **Line (1):**  
   \[
   2x^2 + 5x - 4 > \frac{x^2 - x + 3}{2} 
   \]
   The student multiplies both sides by 2 to eliminate the denominator on the right side:
   \[
   4x^2 + 10x - 8 > x^2 - x + 3
   \]
   This seems correct. No mistake here.

2. **Line (2):**  
   \[
   4x^2 + 10x - 8 > x^2 - x + 3
   \]
   The student subtracts \(x^2 - x + 3\) from both sides:
   \[
   5x^2 + 10x - 8 > -x + 3
   \]
   This is where the mistake occurs. When subtracting \(x^2 - x + 3\) from both sides, the correct expression should be:
   \[
   4x^2 + 10x - 8 - (x^2 - x + 3) = 3x^2 + 11x - 11
   \]
   Instead, the student incorrectly wrote \(5x^2 + 10x - 8 > -x + 3\), which is an error in the subtraction process.  

Thus, **Line (2)** is where the error occurs.

Would you like more details about how to properly simplify these expressions or any further explanation?

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Here are some questions for further exploration:

1. What happens if we solve the correct inequality from the second line?
2. How would the solution differ if the mistake wasn't corrected?
3. How do we handle quadratic inequalities after simplification?
4. Can we verify solutions of quadratic inequalities by graphing?
5. What are the common mistakes when simplifying inequalities?

**Tip:** Always carefully distribute and subtract terms, especially when dealing with expressions that involve variables on both sides of the inequality.

The student simplified a quadratic inequality but made one mistake. On which line did the error occur?

Analyze a quadratic inequality where a mistake occurred in the simplification process. The problem involves algebraic manipulation and subtraction of polynomials, suitable for students in Grades 9-11. Explore the step-by-step solution and learn to avoid common errors.

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