Question

which system of inequalities is represented by the graph?

Answer 1

Answer:

$y\leq 2x+4$

$y\geq -x-6$

Step-by-step explanation:

step 1

Find the equation of the solid line with positive slope

we have the points

(0,4) and (-2,0)

Find the slope

$m=(0-4)/(-2-0)=2$

The equation of the line in slope intercept form is equal to

$y=mx+b$

we have

$m=2\\b=4$

substitute

$y=2x+4$

step 2

Find the equation of the inequality with positive slope

we know that

The solution of the inequality, is the shaded area below the solid line

so

The equation of the inequality is

$y\leq 2x+4$ ----> inequality A

step 3

Find the equation of the line with negative slope

we have the points

(-6,0) and (-4,-2)

Find the slope

$m=(-2-0)/(-4+6)=-1$

The equation of the line in slope intercept form is equal to

$y=mx+b$

we have

$m=2\\point\ (-6,0)$

substitute

$0=-(-6)+b$

solve for b

$0=6+b$

$b=-6$

so

The linear equation is

$y=-x-6$

step 4

Find the equation of the inequality with negative slope

we know that

The solution of the inequality, is the shaded area above the solid line

so

The equation of the inequality is

$y\geq -x-6$ ----> inequality B

therefore

The system of inequalities is

$y\leq 2x+4$ ----> inequality A

$y\geq -x-6$ ----> inequality B