This graph shows the solutions to the inequalities y > =x - 2 and y < 32 - 10. Does the system of inequalities have solutions? If so, which region contains the solutions?

Answer 1

To determine if the system of inequalities \( y \geq x - 2 \) and \( y < 32 - 10 \) has solutions, we first analyze each inequality.

1. Inequality 1: \( y \geq x - 2 \)

   • This is a linear inequality representing the region above (and including) the line \( y = x - 2 \).
   • The line itself has a y-intercept at \(-2\) and a slope of \(1\).

2. Inequality 2: \( y < 32 - 10 \)

   • Simplifying gives \( y < 22 \).
   • This represents the region below the horizontal line \( y = 22 \).

Next, we look for the intersection of these two regions.

• The first inequality includes the area above the line \( y = x - 2 \), while the second inequality restricts the solutions to below the line \( y = 22 \).

To check if the system has solutions, we can find where these two lines intersect:

Set \( x - 2 = 22 \): \[ x - 2 = 22 \implies x = 24 \]

At \( x = 24 \), we find: \[ y = x - 2 \implies y = 24 - 2 = 22 \]

So the lines intersect at the point \( (24, 22) \).

Now we can interpret the system of inequalities:

• The region defined by \( y \geq x - 2 \) is above the line (inclusive).
• The region defined by \( y < 22 \) is below the line (not inclusive).

Thus, the solutions to the system will be the region where \( y \) is at least \( x - 2 \) while also being less than \( 22 \).

Conclusion:

Yes, the system of inequalities has solutions. The solution region is above the line \( y = x - 2 \) and below the line \( y < 22 \). The valid region is bounded by these two lines.

Answer 2

There is a solution, and it is shown by region B.
• B. There is a solution, and it is shown by region C.
C. There is no solution.
D. There is a solution, and it is shown by region A.

Answer 3

To answer your question regarding which region (A, B, or C) represents the solution to the inequalities \( y \geq x - 2 \) and \( y < 22 \):

1. Interpretation of the inequalities:
   • \( y \geq x - 2 \): This inequality includes the area above the line \( y = x - 2 \).
   • \( y < 22 \): This inequality includes the area below the horizontal line \( y = 22 \).

Given this, we can determine:

• The area that satisfies both inequalities will be above the line \( y = x - 2 \) and below the line \( y = 22 \).

2. Understanding Regions: Without the graph itself, the specifics about regions A, B, and C are unclear. However, based on typical descriptions of such regions:
   • The solution should lie in the overlap area between the two sets defined by the inequalities.

Possible Options:

• A: May represent the correct overlapping area.
• B: May represent a non-overlapping area or part of the region that does not satisfy one of the inequalities.
• C: Indicates that there are no solutions, which we know isn't true since overlaps exist.

Conclusion:

If Region A contains the overlapping area satisfying both inequalities, then "D. There is a solution, and it is shown by region A" would be the correct choice.

If you have specific descriptions or graphical representations of which regions correspond to A, B, and C, you can clarify further, and I can provide a more definitive answer.