We need to identify the graphs that solve the following system of inequalities:
[tex]\begin{gathered} y<-x+4\text{ (1)} \\ y\ge\frac{2x}{5}-1\text{ (2)} \end{gathered}[/tex]
The first step to do this is to determine which lines refer to each inequality. They have been labeled as (1) and (2) to help us determine the lines.
The inequality (1) is associated with the following line:
[tex]y=-x+4[/tex]
Where the slope (inclination of the line) is -1 and the y-intercept (the point, where the line touches the y-graph, is 4). With this we can determine that the line that start on the upper right corner of the graph and goes down is the line (1).
The inequality (2) is associated with the following line:
[tex]y=\frac{2}{5}x-1[/tex]
Where the slope is 2/5 and the y-intercept is -1. With this we can determine that the line that represents this equation starts on the bottom left of the graph and goes to the upper right.
Now that we know which line represents each equation we can analyze the solution of the inequalities separetly. Since the first inequality has a "<" sign, we need to mark everything that is below it, as shown below:
The second equation, however, has a ">=" sign. Therefore we need to mark everything above it:
The solution to the equation will be the region that is common to the two graphs above. Therefore the final graph is:
The correct answer is D.
