A system of linear inequalities is given.

Question

Which graph shows the solution to this system of inequalities?

A
.

A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.

B.

A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.

C.

A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.

D.

A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.

Answer 1

To determine which graph represents the solution to the system of linear inequalities, we need to analyze the properties of the lines and the shaded regions.

1. Solid vs. Dashed Lines:

   • Solid lines indicate that the points on the line are included in the solution (i.e., the inequalities are "less than or equal to" or "greater than or equal to").
   • Dashed lines indicate that the points on the line are not included in the solution (i.e., the inequalities are "less than" or "greater than").

2. Shaded Regions:

   • The area shaded below the line typically represents inequalities of the form \(y < mx + b\) (for upper bounding lines).
   • Conversely, the area shaded above the line represents inequalities of the form \(y > mx + b\) (for lower bounding lines).

Now, let’s analyze the scenarios:

• Graph A has solid lines with shaded regions below. This means the inequalities are of the form \(\leq\) and \(<\).
• Graph B has dashed lines with shaded regions below, indicating the inequalities are strictly less than.
• Graph C has solid lines with shaded regions above, indicating the inequalities might be of the form \(>\).
• Graph D has dashed lines with shaded regions above, indicating the inequalities are strictly greater than.

To select the correct graph based on this analysis, you would need the specific inequalities to see which one fits the situation described above. However, based solely on the nature of the lines and shading described, if the inequalities were:

• solid lines indicating inclusion (this would favor Graph A or C based on how shading is oriented).
• and the common region was below or above those lines depending on the exact inequalities.

Without specifications on the inequalities, I cannot definitively choose one, but if the lines are supposed to be inclusive and bound a region below, Graph A would generally represent a solution where the inequalities allow for inclusion below both lines. If the inequalities are strict, then Graph B could be valid.