Answer:
D. [tex]y<\frac{2}{3}x-4\ and\ y\geq-2x+2[/tex]
Step-by-step explanation:
Given:
Let us find the equations of the lines from the graph.
First let us determine the equation of the broken line.
The slope of the broken line is positive as 'y' values increases with increase in 'x'. The slope is the ratio of the absolute value of y-intercept to that of the x-intercept.
x-intercept = 6, |y-intercept| = |-4| = 4
So, [tex]m=\frac{4}{6}=\frac{2}{3}[/tex]
Now, equation of a line with slope 'm' and y -intercept 'b' is given as:
[tex]y=mx+b[/tex]
Here, [tex]m=\frac{2}{3},b=-4[/tex].So, equation of the broken line is:
[tex]y=\frac{2}{3}x-4[/tex]
Now, from the graph, the solution is below the broken line. So, the equality sign is replaced by the less than inequality sign. So,
[tex]y<\frac{2}{3}x-4[/tex]
Now, let us determine the equation of the other line.
y-intercept, [tex]b = 2[/tex], x-intercept = 1
Slope is negative as 'y' decreases with increase in 'x'. So,
Slope, [tex]m=-\frac{2}{1}=-2[/tex]
Now, equation is given as:
[tex]y=-2x+2[/tex]
From the graph, the solution region is to the left of the line. So, the 'equal to' sign is replaced by the 'greater than or equal to' sign' as the line is also included in the solution region. So, the inequality becomes:
[tex]y\geq-2x+2[/tex]
Therefore, the last option is correct.
[tex]y<\frac{2}{3}x-4\ and\ y\geq-2x+2[/tex]
