Question

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:

y < 4x − 8
Y>-5/2x+5
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 (5, −8) included in the solution area for the system? Justify your answer mathematically.

Answer 1

The graph features dashed and solid lines, shading below one line and above the other, forming an overlapping bounded area. Point (5, −8) falls within the solution area, meeting both inequalities.

To graph the system of inequalities y < 4x - 8 and
`\(y > -(5)/(2)x + 5\)`, we'll start by plotting the lines representing each inequality on the coordinate plane.

Graph Description:

Lines:

y = 4x - 8 (dashed line)

`\(y = -(5)/(2)x + 5\)` (solid line)

Shading**:

For y < 4x - 8, the area below the line y = 4x - 8 is shaded.

For
`\(y > -(5)/(2)x + 5\)`, the area above the line
`\(y = -(5)/(2)x + 5\)` is shaded.

Solution Area:

The solution area is the region that satisfies both inequalities, i.e., the overlapping shaded region.

Solution:

Upon graphing both inequalities and analyzing their respective shaded regions, the solution area is the region below the line y = 4x - 8 and above the line
`\(y = -(5)/(2)x + 5\)`. These areas overlap, creating a bounded region between the lines.

Point (5, −8) Inclusion:

Let's check if the point (5, −8) lies within the solution area by substituting its coordinates into the inequalities:

1. For y < 4x - 8:

Substituting (5, −8):

-8 < 4(5) - 8

-8 < 20 - 8

-8 < 12 (True)

2. For
`\(y > -(5)/(2)x + 5\`):

Substituting (5, −8):

`\(-8 > -(5)/(2)(5) + 5\)`

`\(-8 > -(25)/(2) + 5\)`

`\(-8 > -(15)/(2)\)` (True)

Conclusion:

The point (5, −8) satisfies both inequalities, falling within the solution area for the system of inequalities. Mathematically, it belongs to the shaded region that satisfies both y < 4x - 8 and
`\(y > -(5)/(2)x + 5\)` on the graphed plane.