Answer :
To determine which graph represents the given system of inequalities, we need to analyze each inequality and then combine their effects on the coordinate plane.
The given system of inequalities is:
[tex]\[ \begin{array}{l} y \geq 3x + 1 \\ x > -3 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the First Inequality: \( y \geq 3x + 1 \)
- First, graph the line \( y = 3x + 1 \). This is a linear equation, so it will form a straight line.
- Find the y-intercept (where \( x = 0 \)):
[tex]\[ y = 3(0) + 1 = 1 \][/tex]
So, the line crosses the y-axis at (0, 1).
- Find another point using \( x = 1 \):
[tex]\[ y = 3(1) + 1 = 4 \][/tex]
So, the point (1, 4) lies on the line.
- Plot these points and draw the line through them.
- Since the inequality is \( y \geq 3x + 1 \), we shade the region above the line (including the line itself, as it's a "greater than or equal to" inequality).
2. Graph the Second Inequality: \( x > -3 \)
- This inequality represents a vertical line at \( x = -3 \).
- Draw a dashed vertical line at \( x = -3 \) since it is a strict inequality (not including the line itself).
- Shade the region to the right of this line, as the inequality suggests \( x \) must be greater than -3.
3. Combine the Two Inequalities:
- The solution to the system of inequalities will be the region where the shaded areas from both inequalities overlap.
- Ensure to identify the region that satisfies both \( y \geq 3x + 1 \) and \( x > -3 \).
### The Resulting Graph:
1. Line \( y = 3x + 1 \):
- Solid line (included in the solution).
- Runs through (0, 1) and (1, 4).
- Above this line is shaded.
2. Vertical dashed line at \( x = -3 \):
- Excluded in the solution.
- Region to the right of this line is shaded.
The overlapping region will be the solution to the system:
- It is the area above the line \( y = 3x + 1 \).
- It is also the area to the right of the line \( x = -3 \).
Thus, the graph that represents the system will show a shaded region above the line \( y = 3x + 1 \), starting from \( x > -3 \) and extending infinitely to the right.
This graphical representation will ensure we capture the appropriate solution region for the given system of inequalities.
The given system of inequalities is:
[tex]\[ \begin{array}{l} y \geq 3x + 1 \\ x > -3 \end{array} \][/tex]
### Step-by-Step Solution:
1. Graph the First Inequality: \( y \geq 3x + 1 \)
- First, graph the line \( y = 3x + 1 \). This is a linear equation, so it will form a straight line.
- Find the y-intercept (where \( x = 0 \)):
[tex]\[ y = 3(0) + 1 = 1 \][/tex]
So, the line crosses the y-axis at (0, 1).
- Find another point using \( x = 1 \):
[tex]\[ y = 3(1) + 1 = 4 \][/tex]
So, the point (1, 4) lies on the line.
- Plot these points and draw the line through them.
- Since the inequality is \( y \geq 3x + 1 \), we shade the region above the line (including the line itself, as it's a "greater than or equal to" inequality).
2. Graph the Second Inequality: \( x > -3 \)
- This inequality represents a vertical line at \( x = -3 \).
- Draw a dashed vertical line at \( x = -3 \) since it is a strict inequality (not including the line itself).
- Shade the region to the right of this line, as the inequality suggests \( x \) must be greater than -3.
3. Combine the Two Inequalities:
- The solution to the system of inequalities will be the region where the shaded areas from both inequalities overlap.
- Ensure to identify the region that satisfies both \( y \geq 3x + 1 \) and \( x > -3 \).
### The Resulting Graph:
1. Line \( y = 3x + 1 \):
- Solid line (included in the solution).
- Runs through (0, 1) and (1, 4).
- Above this line is shaded.
2. Vertical dashed line at \( x = -3 \):
- Excluded in the solution.
- Region to the right of this line is shaded.
The overlapping region will be the solution to the system:
- It is the area above the line \( y = 3x + 1 \).
- It is also the area to the right of the line \( x = -3 \).
Thus, the graph that represents the system will show a shaded region above the line \( y = 3x + 1 \), starting from \( x > -3 \) and extending infinitely to the right.
This graphical representation will ensure we capture the appropriate solution region for the given system of inequalities.
