To determine which graph corresponds to the given system of inequalities:

1. Inequality Analysis:

   • $y \geq -\frac{3}{4}x + 4$
     • This represents a line with a slope of $-\frac{3}{4}$ and a y-intercept of $4$. The inequality indicates that the region above (or on) the line is shaded.
   • $y < \frac{1}{2}x - 1$
     • This line has a slope of $\frac{1}{2}$ and a y-intercept of $-1$. The inequality indicates that the region below (but not including) the line is shaded.

2. Graph Characteristics:

   • The line corresponding to $y \geq -\frac{3}{4}x + 4$ should be solid since the inequality includes equality ($\geq$).
   • The line for $y < \frac{1}{2}x - 1$ should be dashed, as the inequality does not include equality ($<$).

3. Shading:

   • For the first inequality ($y \geq -\frac{3}{4}x + 4$), the area above the line should be shaded.
   • For the second inequality ($y < \frac{1}{2}x - 1$), the area below the line should be shaded.

Now, let's match these criteria with the graphs shown:

• The correct graph should have:
  • A solid line with a slope of $-\frac{3}{4}$ and y-intercept $4$, with the area above it shaded.
  • A dashed line with a slope of $\frac{1}{2}$ and y-intercept $-1$, with the area below it shaded.

Upon analyzing the provided graphs, the graph that meets all these conditions is the last graph (bottom option) in the images.

Would you like further details on how these inequalities are graphed or have any other questions?

Related Questions:

1. What is the method to find the intersection point of two lines?
2. How do you determine which side of the line to shade for an inequality?
3. Why do some lines appear dashed while others are solid in graphs of inequalities?
4. How does the slope of a line affect the direction it goes on the graph?
5. What happens if the inequalities are switched to $>$ or $\leq$?

Tip:

Always start by identifying the slope and y-intercept of each line when analyzing or graphing inequalities. This makes it easier to set up the graph correctly.