Lola graphed the system of inequalities shown here.

 
y > 3x − 52
y < −12x +1




Select the section of the graph that represents the solution to the system of inequalities.

To find the section of the graph that represents the solution to this system of inequalities, we’ll analyze the inequalities separately and then identify where their solutions overlap.

### Step 1: Understand Each Inequality

1. **Inequality 1:** \( y > 3x - 52 \)
   - This inequality represents a line with slope \( 3 \) and \( y \)-intercept \( -52 \). 
   - Since the inequality is \( y > 3x - 52 \), the solution includes the region **above** this line.

2. **Inequality 2:** \( y < -12x + 1 \)
   - This inequality represents a line with slope \( -12 \) and \( y \)-intercept \( 1 \).
   - Since the inequality is \( y < -12x + 1 \), the solution includes the region **below** this line.

### Step 2: Identify the Solution Region

The solution to the system is the region where **both conditions are satisfied simultaneously**:
- Above the line \( y = 3x - 52 \).
- Below the line \( y = -12x + 1 \).

### Step 3: Graph Interpretation

On the graph, the solution region will be the overlapping shaded area that lies:
- Above the \( y = 3x - 52 \) line, and
- Below the \( y = -12x + 1 \) line.

If you're looking at the graph, select the section of the plane where these two conditions intersect. 

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Would you like more details or have any questions on this? 

Here are some questions to explore further:

1. How do we determine which side of a line represents the inequality solution?
2. Why does the intersection region represent the solution for the system of inequalities?
3. How would the solution change if one of the inequalities used \( \geq \) or \( \leq \) instead?
4. How can we verify the solution by picking test points?
5. What happens to the solution region if the slopes of the lines are closer in value?

**Tip:** When solving systems of inequalities, it’s often helpful to shade the regions separately and then look for the overlap to avoid errors in interpretation.

Lola graphed the system of inequalities shown here. y > 3x − 52, y < −12x +1. Select the section of the graph that represents the solution to the system of inequalities.

Explore how to graph the system of inequalities y > 3x - 52 and y < -12x + 1, identifying the overlapping solution region. This problem focuses on graphing and interpreting linear inequalities, suitable for Grades 9-10.

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