Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.