Answer:
D
Step-by-step explanation:
To find the system of inequalities, we simply need to find the equation of each line.
Thus, let's find the inequality that is represented by each line.
Red Line:
First, let’s determine the slope. To do so, we can use the slope formula:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
We can use any two points that's on the Red Line. Using (0, 1) and (1, 3) and substituting them into the slope formula yields:
[tex]\displaystyle m=\frac{3-1}{1-0}=\frac{2}{1}=2[/tex]
Therefore, the slope of the red line is two.
To write the equation of the red line, we can use the slope-intercept form:
[tex]y=mx+b[/tex]
Notice that our y-intercept of our red line is (0, 1). Therefore, b = 1 and we already determind that m = 2:
[tex]y=2x+1[/tex]
Now, we need to determine our sign. Notice that the shaded region is below the red line. Therefore, y is less than our equation.
And since our line is shaded, our sign will be less than or equal to. Therefore:
[tex]y\leq 2x+1[/tex]
Blue Line:
Again, let’s first find the slope. We can use the two points (0, 2) and (2, 3). Hence:
[tex]\displaystyle m=\frac{3-2}{2-0}=\frac{1}{2}[/tex]
So, the slope is 1/2.
We can now use the slope-intercept form. Notice that our y-intercept is (0, 2). Thus, we will substitute 1/2 for m and 2 for b. This yields:
[tex]\displaystyle y = \frac{1}{2}x+2[/tex]
Finally, we will determine our sign. Since the shaded region is above the blue line, our y is greater than our equation. And since it’s shaded, our sign is greater than or equal to. Therefore, our equation is:
[tex]\displaystyle y\geq\frac{1}{2}x+2[/tex]
Thus, all together, our inequalities are:
[tex]\displaystyle y\leq 2x+1\\ \\ y\geq \frac{1}{2}x+2[/tex]
Therefore, our answer is D
