Question

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions: y < 2x − 7 y is greater than or equal to negative 1 over 2 times x plus 3
Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution area.

Part B: Is the point (3, −7) included in the solution area for the system? Justify your answer mathematically.

Answer 1

Solution-

The given inequalities are,

$y < 2x-7,\\\\y\geq-\frac{1}{2}x+3$

Part A

As the former inequality is only '<' so it will be represented as dotted line, but in case of the later inequality there is '≥' so it will be represented as a solid line.

This happens because in only '<, >' the values must be either lesser or greater than y, they can't take the values of y, but in case of '≤, ≥' the values can be equal to y.

Taking origin (0, 0) and putting in the first inequality.

$\Rightarrow 0< 2(0)-7\\\\\Rightarrow 0\nless -7$

As the equation does not satisfy, so the shaded area will be on the other side of the origin.

Again taking origin (0, 0) and putting in the second inequality.

$\Rightarrow 0\geq-\frac{1}{2}0+3\\\\\Rightarrow 0\ngeqslant 3$

As the equation does not satisfy, so the shaded area will be on the other side of the origin.

The second attachment contains the feasible region.

Part B

In order to know whether the point (3, −7) is included in the solution area for the system or not, we must put it in both the equations, if it satisfies both then only it is within the solution area.

$\Rightarrow -7< 2(3)-7,\ -7\geq-\frac{1}{2}(3)+3\\\\\Rightarrow -7$

As the point (3, -7) only satisfies first inequality, but not the second inequality so this point won't be in the solution area.

Answer 2

Answer:

Part A: The graph of given system of inequalities shown below.

Part B: (3,-7) is not included in the solution area.

Step-by-step explanation:

The given system of equations is

$y$         .... (1)

$y\ge -\frac{1}{2}x+3$         ... (2)

Part A:

The related equations are

$y=2x-7$               .... (3)

$y= -\frac{1}{2}x+3$        ... (4)

Table of values

For equation 3.

x      y

0     -7

3.5   0

For equation 4.

x      y

0     3

6     0

Plot these ordered pairs on a coordinate planes and connect corresponding lines to draw the straight lines.

Check each inequality by (0,0).

$(0)$  (False)

$(0)\ge -\frac{1}{2}(0)+3\Rightarrow 0\geq 3$    (False)

It means (0,0) is not included in the shaded region of any of these inequalities.

$y$ is dotted line.

$y\ge -\frac{1}{2}x+3$ is a solid line.

Therefore, the graph of given system of inequalities shown below.

Part B:

Check each each inequality by (3,-7).

$(-7)$       (True)

$(-7)\ge -\frac{1}{2}(3)+3\Rightarrow -7\ge 1.5$     (False)

Since second inequality is not satisfied by the point (3,-7), therefore (3,-7) is not included in the solution area.