Respuesta :

Sushi121212
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.
calculista

Step 1

Find the slope of the dotted line A

Let

[tex]A(-3,1)\\B(0,2)[/tex]

Find the slope m

[tex]m=\frac{(y2-y1)}{(x2-x1)}[/tex]

substitute in the formula

[tex]m=\frac{(2-1)}{(0+3)}[/tex]

[tex]m=\frac{(1)}{(3)}[/tex]

[tex]mA=\frac{1}{3}[/tex]

Step 2

Find the equation of the line A

we know that

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]mA=\frac{1}{3}[/tex]

[tex]B(0,2)[/tex]

substitute

[tex]y-2=\frac{1}{3}(x-0)[/tex]

[tex]y=\frac{1}{3}x+2[/tex]

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    

[tex]y<\frac{1}{3}x+2[/tex]

Step 4

Find the slope of the solid line B

Let

[tex]B(0,2)\\C(1,0)[/tex]

Find the slope m

[tex]m=\frac{(y2-y1)}{(x2-x1)}[/tex]

substitute in the formula

[tex]m=\frac{(0-2)}{(1-0)}[/tex]

[tex]m=\frac{(-2)}{(1)}[/tex]

[tex]mB=-2[/tex]

Step 5

Find the equation of the line B

we know that

[tex]y-y1=m(x-x1)[/tex]

we have

[tex]mB=-2[/tex]

[tex]B(0,2)[/tex]

substitute

[tex]y-2=-2(x-0)[/tex]

[tex]y=-2x+2[/tex]

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    

[tex]y\leq -2x+2[/tex]

therefore

The system of linear inequalities is

[tex]y<\frac{1}{3}x+2[/tex]

[tex]y\leq -2x+2[/tex]

using a graphing tool

the solution is the shaded area

see the attached figure


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