Question

On a piece of paper, graph this system of inequalities. Then determine which region contains the solution to the system.

Answer 1

we have

$y\leq \frac{1}{4}x+3$ ----> inequality 1

The solution of the inequality 1 is the shaded area below the solid red line

The solution is the region D and region C

$y\geq-x+5$ -----> inequality 2

The solution of the inequality 2 is the shaded area above the solid blue line

The solution is the region B and region C

The solution of the system is the common area

so

The solution is the region C

see the attached figure

therefore

the answer is the option C

Region C

Answer 2

Answer:

Region C of  the graph will contains the solution to the given system

Step-by-step explanation:

 Consider the given system of equation

$y\le\frac{1}{4}x+3$

and $y\ge-x+5$

We have to choose the region of the graph that contains the solution to the given system.

Since, to determine the region choose  a test point in each region and then then check the values of inequality at that point and for the test point that satisfies both the inequality  will contains the solution to the given system.

On region A)

Let (0, 4)  be the test point that lies in region A

Then put the value of x = 0 and y= 4 in given system,

we have,

$4\le\frac{1}{4}(0)+3 \Rightarrow 4\le 3$ (false)

and $4\ge(0)+5 \Rightarrow 4\ge 5$ (false)

On region B)

Let (0, 8)  be the test point that lies in region B

Then put the value of x = 0 and y= 8 in given system,

we have,

$8\le\frac{1}{4}(0)+3 \Rightarrow 8\le 3$ (false)

and $8\ge(0)+5 \Rightarrow 8\ge 5$ (true)

On region C)

Let (8,0)  be the test point that lies in region C

Then put the value of x = 8 and y= 0 in given system,

we have,

$0\le\frac{1}{4}(8)+3 \Rightarrow 0\le 5$ (true)

and $0\ge(-8)+5 \Rightarrow 0\ge -3$ (true)

On region D)

Let (0,0)  be the test point that lies in region D

Then put the value of x = 0 and y= 0 in given system,

we have,

$0\le\frac{1}{4}(0)+3 \Rightarrow 0\le 3$ (false)

and $0\ge(0)+5 \Rightarrow 0\ge 5$ (false)

Thus, only (8,0)  be the test point that lies in region C satisfies both inequality.

Thus, region C of  the graph will contains the solution to the given system