Answer:
A and B look identical, so perhaps there was a typo?
The equations are:
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:
Let (0, 3) be (x₁, y₁) and let's let (3, 4) be (x₂, y₂). So, our slope is:
Subtract:
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:
Where m is the slope and b is the y-intercept.
Substitute 1/3 for m and 3 for b. This yields:
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:
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:
Evaluate:
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:
Substitute 3 for m and -2 for b:
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:
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:
The first term is traditionally positive. So, multiply both sides by -1. Since we're multiplying by a negative, flip the inequality sign:
We can see that the answer choice that matches this is A (or B?)
And we're done!