Question

What system of linear inequalities is shown in the graph? Enter your answers in the boxes.

Answer 1

Step 1

Find the slope of the dotted line A

Let

`A(-3,1)\\B(0,2)`

Find the slope m

`m=((y2-y1))/((x2-x1))`

substitute in the formula

`m=((2-1))/((0+3))`

`m=((1))/((3))`

`mA=(1)/(3)`

Step 2

Find the equation of the line A

we know that

`y-y1=m(x-x1)`

we have

`mA=(1)/(3)`

`B(0,2)`

substitute

`y-2=(1)/(3)(x-0)`

`y=(1)/(3)x+2`

Step 3

Find the equation of the inequality A

we know that

the solution is below the dotted line A

so

the inequality A is

`y<(1)/(3)x+2`

Step 4

Find the slope of the solid line B

Let

`B(0,2)\\C(1,0)`

Find the slope m

`m=((y2-y1))/((x2-x1))`

substitute in the formula

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

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

`mB=-2`

Step 5

Find the equation of the line B

we know that

`y-y1=m(x-x1)`

we have

`mB=-2`

`B(0,2)`

substitute

`y-2=-2(x-0)`

`y=-2x+2`

Step 6

Find the equation of the inequality B

we know that

the solution is below the solid line B

so

the inequality B is

`y\leq -2x+2`

therefore

The system of linear inequalities is

`y<(1)/(3)x+2`

`y\leq -2x+2`

using a graphing tool

the solution is the shaded area

see the attached figure

Answer 2

First, find the equation of the lines:

y=-2x+2 and y=1/3x+2

We see that the shaded part is below the y=1/3x+2 graph, so it becomes y<1/3x+2 (It's not ≤ since it's a dotted line)

For the y=-2x+2, the shaded part is under the line as well, so it becomes y≤-2x+2 (Here, we include the ≤ since it's a solid line)

So your answer is y<1/3x+2 and y≤-2x+2.