https://file.aiquickdraw.com/mathbot/file/1726978756719kqvv3vg0.jpg

The system of inequalities given in the image is:

1. \( 5x - 4y < 4 \)
2. \( x + y < 2 \)

To determine which graph represents the solution to these inequalities, let’s analyze both inequalities.

### First Inequality: \( 5x - 4y < 4 \)

1. Rearrange the inequality into slope-intercept form (\( y = mx + b \)):
   \[
   5x - 4y < 4 \quad \Rightarrow \quad -4y < -5x + 4 \quad \Rightarrow \quad y > \frac{5}{4}x - 1
   \]
   So, the boundary line is \( y = \frac{5}{4}x - 1 \), but since the inequality is \( > \), you will shade **above** the line, and the line will be dashed.

### Second Inequality: \( x + y < 2 \)

1. Rearrange this into slope-intercept form:
   \[
   x + y < 2 \quad \Rightarrow \quad y < -x + 2
   \]
   The boundary line is \( y = -x + 2 \), and since the inequality is \( < \), you will shade **below** the line, with a dashed boundary.

### Graphing

- The first line (\( y = \frac{5}{4}x - 1 \)) is sloped upward, with shading above it.
- The second line (\( y = -x + 2 \)) slopes downward, with shading below it.
- The solution region is the overlap of the shaded areas, where both inequalities hold true.

If you were to graph these inequalities, you'd use dashed lines for both boundaries since the inequalities do not include equality (\( < \), not \( \leq \)).

Do you need more details on solving or graphing these inequalities? Also, here are some relative questions you might find useful:

1. What is the significance of using dashed lines in inequality graphs?
2. How would the solution change if the inequalities were \( \leq \) instead of \( < \)?
3. What is the general method to graph a linear inequality?
4. Can you describe the graphical difference between \( y > \) and \( y < \)?
5. How do you identify the region where two inequalities overlap?

**Tip:** Always check whether the inequality involves a strict \( < \) or \( \leq \), as it determines whether the boundary line is solid or dashed in a graph.

Which graph represents the solution to the system of inequalities: 5x - 4y < 4 and x + y < 2?

Explore how to graph the system of linear inequalities 5x - 4y < 4 and x + y < 2. This problem involves converting inequalities into slope-intercept form, determining the shading regions, and finding the solution set's overlap. Ideal for high school students in Grades 9-11 learning algebra and graphing concepts.

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