Lola graphed the system of inequalities shown here.

 

y > 3x − 52y > 3x − 52

y < −12x +1y < −12x +1





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

To find the solution to this system of inequalities, let's analyze each inequality individually and identify where their solution regions overlap on the graph:

### Inequality Breakdown
1. **First Inequality**: \( y > 3x - 5 \)
   - This represents a region above the line \( y = 3x - 5 \), which has a slope of 3 and a y-intercept at -5.
   - Since it's a strict inequality ("\(>\)"), the boundary line \( y = 3x - 5 \) is dashed, indicating points on the line are *not* included in the solution.

2. **Second Inequality**: \( y < -12x + 1 \)
   - This represents a region below the line \( y = -12x + 1 \), which has a slope of -12 and a y-intercept at 1.
   - Being a strict inequality ("\( < \)"), the boundary line \( y = -12x + 1 \) is also dashed, meaning points on this line are *not* included.

### Graphing the Solution
To graph the solution:
1. Draw both lines \( y = 3x - 5 \) and \( y = -12x + 1 \) as dashed lines.
2. Shade the region **above** \( y = 3x - 5 \).
3. Shade the region **below** \( y = -12x + 1 \).

### Finding the Overlapping Region
The solution to the system is where the shaded regions from each inequality overlap on the graph. This overlapping region is the answer, representing all the points that satisfy **both** inequalities.

If you have the graph, look for the section where these two shaded areas intersect and select that area as the solution to the system.

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Would you like a detailed example of shading or a graphical representation? Here are some related questions for further practice:

1. How would the graph change if one inequality was non-strict (e.g., \( \geq \) or \( \leq \))?
2. What does the slope of each line tell you about the direction of shading?
3. How would the solution region change if one line had a positive y-intercept?
4. Can a system of inequalities have no solution region? Why or why not?
5. How would you test a point to see if it lies in the solution region?

**Tip:** When working with inequalities, always check where the regions intersect, as this determines the solution to the system.

Lola graphed the system of inequalities shown here: y > 3x - 5 and y < -12x + 1. Select the section of the graph that represents the solution to the system of inequalities.

Learn how to solve and graph the system of inequalities y > 3x - 5 and y < -12x + 1. This lesson covers key concepts in graphing linear inequalities and finding overlapping solution regions. Suitable for Grades 8-10, this example explains shading techniques and region identification.

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