Answer:
A and B look identical, so perhaps there was a typo?
The equations are:
![y\geq1/3x+3](https://tex.z-dn.net/?f=y%5Cgeq1%2F3x%2B3)
![3x-y>2](https://tex.z-dn.net/?f=3x-y%3E2)
Step-by-step explanation:
To find the system of linear inequalities represented by the graph, we need to find the equation of each line.
Red Line:
To find the equation of the red line, we need to first find the slope and the y-intercept.
Let's pick two points from the red line. We can use (0, 3) and (3, 4).
The slope formula is:
![m=\frac{y_2-y_1}{x_2-x_1}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D)
Let (0, 3) be (x₁, y₁) and let's let (3, 4) be (x₂, y₂). So, our slope is:
![m=\frac{4-3}{3-0}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B4-3%7D%7B3-0%7D)
Subtract:
![m=1/3](https://tex.z-dn.net/?f=m%3D1%2F3)
So, our slope is 1/3.
Note that (0,3) is also our y-intercept. So, our y-intercept is y=3.
Now, we can use the slope-intercept form:
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
Where m is the slope and b is the y-intercept.
Substitute 1/3 for m and 3 for b. This yields:
![y=1/3x+3](https://tex.z-dn.net/?f=y%3D1%2F3x%2B3)
Now, we need to determine which sign to use.
Note that the red line is <em>not</em> dotted, meaning we need "or equal to."
Also, note that the shaded region is <em>above</em> our line. In other words, y must be <em>greater</em> than our equation.
So, our sign is "greater than or equal to."
This, the red inequality is represented by the equation:
![y\geq1/3x+3](https://tex.z-dn.net/?f=y%5Cgeq1%2F3x%2B3)
Blue Line:
Again, let's first find the slope and the y-intercept.
We can use the points (0, -2) and (1,1). Let (0,-2) be (x₁, y₁) and let's let (1, 1) be (x₂, y₂). Substitute them into the slope formula:
![m=\frac{1-(-2)}{1-0}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B1-%28-2%29%7D%7B1-0%7D)
Evaluate:
![m=3/1=3](https://tex.z-dn.net/?f=m%3D3%2F1%3D3)
So, our slope is 3.
(0, -2) is also our y-intercept. So, our y-intercept is y=-2.
Again, we can use the slope-intercept form:
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
Substitute 3 for m and -2 for b:
![y=3x-2](https://tex.z-dn.net/?f=y%3D3x-2)
Now, we need to find our inequality sign.
Notice that the line <em>is</em> dotted. So, we won't have any "or equal to."
Also, notice that our shaded region is <em>below</em> our line. Thus, y is <em>less than</em> our equation. Therefore:
![y](https://tex.z-dn.net/?f=y%3C3x-2)
However, all of our choices are in standard form. So, we need to convert this to standard form.
To do so, subtract 3x from both sides:
![-3x+y](https://tex.z-dn.net/?f=-3x%2By%3C-2)
The first term is traditionally positive. So, multiply both sides by -1. Since we're multiplying by a negative, flip the inequality sign:
![3x-y>2](https://tex.z-dn.net/?f=3x-y%3E2)
We can see that the answer choice that matches this is A (or B?)
And we're done!