Question

What system of linear inequalities is represented by the graph?

Answer 1

first line:
y=0.5x-1

second/dotted line:
y=2x-4

your allowed values are below the first line and above the second, so the system is
I: y<=0.5x-1
II: y>=2x-4

your dotted line might also mean > instead of >=, but this depends on the notation used in class

Answer 2

y > 2x - 4 and y ≤
`(x)/(2)-1`

Explanation:

In this question we will find the equations of the lines first then decide the inequality sign.

Dotted line in the graph passes through two points (2, 0) and (0, -4)

Let the equation of the line is y = mx + c

Where m = slope and c = y-intercept

Slope (m) of the line will be =
`(y-y')/(x-x')`

m =
`(0+4)/(2-0)`

m = 2

y-intercept of the line = (-4)

So, equation of the line will be y = 2x - 4

Now we see the shaded area is above the dotted line so inequality will be

y > 2x - 4

Now for the bold line

Let the equation of this line is y = m'x + c'

This line passes through points (2, 0) and (0, -1)

then slope of the line m' =
`(y-y')/(x-x')`

m =
`(0+1)/(2-0)` =
`(1)/(2)`

y-intercept = c = -1

Then equation of the line will be

y =
`(x)/(2)-1`

Since shaded area is below the bold line therefore, inequality sign will be "less than equal to"

Inequality of the bold line will be y ≤
`(x)/(2)-1`

System of the linear inequalities will be

y > 2x - 4 and y ≤
`(x)/(2)-1`