Question

Is this a graph of a system of inequalities with no solutions?: y > x + 2, y < x - 2

Answer 1

To determine whether the system of inequalities has solutions, let's analyze the inequalities:

1. \( y > x + 2 \): This represents a region above the line \( y = x + 2 \).
2. \( y < x - 2 \): This represents a region below the line \( y = x - 2 \).

Next, let's find the points where these two lines intersect to see if there's an overlap in the regions defined by the inequalities.

**Finding the intersection points:**

Set the two equations equal to each other:
\[
x + 2 = x - 2
\]

Subtract \( x \) from both sides:
\[
2 = -2
\]

This is a contradiction, meaning that the lines \( y = x + 2 \) and \( y = x - 2 \) are parallel and will never intersect.

**Analyzing the regions:**

- The area described by \( y > x + 2 \) is above the line \( y = x + 2 \).
- The area described by \( y < x - 2 \) is below the line \( y = x - 2 \).

Since one region is above and the other is below, there cannot be any points that satisfy both inequalities simultaneously.

**Conclusion:**
Yes, this system of inequalities has no solutions, as the two regions do not overlap at all.