To analyze the system of inequalities:
- \( y > \frac{1}{2}x + 5 \)
- \( y < -2x + 1 \)
we can graph them and find the solution set.
Step 1: Graph the first inequality \( y > \frac{1}{2}x + 5 \)
- The line \( y = \frac{1}{2}x + 5 \) is a straight line with a slope of \( \frac{1}{2} \) and a y-intercept of 5.
- Since the inequality is \( > \), we will draw a dashed line (indicating that points on the line are not included in the solution) and shade above the line.
Step 2: Graph the second inequality \( y < -2x + 1 \)
- The line \( y = -2x + 1 \) has a slope of -2 and a y-intercept of 1.
- Since the inequality is \( < \), we will again draw a dashed line and shade below the line.
Step 3: Identify the solution region
The solution region is where the shading from both inequalities overlaps. This area represents all the (x, y) pairs that satisfy both inequalities.
Finding Intersection Points
To sketch these graphs accurately, it's useful to find the points where these two lines intersect.
Set the equations equal to find the intersection:
\[ \frac{1}{2}x + 5 = -2x + 1 \]
Multiply everything by 2 to eliminate the fraction:
\[ x + 10 = -4x + 2 \]
Combine like terms:
\[ 5x = -8 \]
Solve for \( x \):
\[ x = -\frac{8}{5} \quad \text{or} \quad -1.6 \]
Now, substitute \( x = -\frac{8}{5} \) back into one of the equations to find \( y \):
\[ y = \frac{1}{2} \left(-\frac{8}{5}\right) + 5 = -\frac{4}{5} + 5 = \frac{21}{5} \quad \text{(or 4.2)} \]
So the point of intersection is: \[ \left(-\frac{8}{5}, \frac{21}{5}\right) \quad \text{or approximately } (-1.6, 4.2) \]
Conclusion
To find the solution set:
- Graph both lines,
- Shade appropriately,
- The overlapping shaded region is where \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \) are both satisfied.
This graphical representation will help visualize the solution. If you need exact numerical solutions for specific values or additional details, feel free to ask!