Question

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions: y < 4x − 2 y is greater than or equal to negative 5 over 2 times x minus 2 Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution area. (6 points) Part B: Is the point (−2, −2) included in the solution area for the system? Justify your answer mathematically. (4 points)

Answer 1

See explanation

Explanation:

You have to graph the system of inequalities presented as

`\left\{\begin{array}{l}y<4x-2\\y\ge -(5)/(2)x-2\end{array}\right.`

Part A:

1. Draw a dotted line
`y=4x-2` (dotted because the sign < is without notion "or equal to"). Select one of two regions by substituting the coordinates of origin into inequality:

`0<4\cdot 0-2\\ \\0<-2`

Since this inequlity is false, the origin doesn't belong to the shaded region, so you have to shade that part which doesn't contain origin (red part in attached diagram).

2. Draw a solid line
`y=-(5)/(2)x-2` (solid because the sign ≥ is with notion "or equal to"). Select one of two regions by substituting the coordinates of origin into inequality:

`0\ge -(5)/(2)\cdot 0-2\\ \\0\ge -2`

Since this inequlity is true, the origin belongs to the shaded region, so you have to shade that part which contains origin (blue part in attached diagram).

The intersection of these two regions is the solution area.

Part B:

Plot point (-2,-2). Since this point doesn't belong to the solution area, this is not a solution of the system of two inequalities. You can check it mathematically - substitute x=-2 and y=-2 into the system:

`\left\{\begin{array}{l}-2<4\cdot (-2)-2\\-2\ge -(5)/(2)\cdot (-2)-2\end{array}\right.\Rightarrow \left\{\begin{array}{l}-2<-10\\-2\ge 3\end{array}\right.`

Both inequalities are false, so (-2,-2) doesn't belong to the solution area.