Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure, let's solve the inequality [tex]\( y + 2 > -3x - 3 \)[/tex] step by step, and then determine what the graph should look like.
1. Rewrite the Inequality:
To put the equation in a more familiar form, we need to isolate [tex]\( y \)[/tex].
[tex]\[ y + 2 > -3x - 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ y > -3x - 3 - 2 \][/tex]
Simplify the right-hand side:
[tex]\[ y > -3x - 5 \][/tex]
2. Identify the Line:
The inequality [tex]\( y > -3x - 5 \)[/tex] involves a linear expression on the right-hand side. To graph this inequality, first, consider the related equation where the inequality is replaced by an equality:
[tex]\[ y = -3x - 5 \][/tex]
3. Graph the Line [tex]\( y = -3x - 5 \)[/tex]:
- This line has a slope of [tex]\(-3\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- To graph it, start at [tex]\((0, -5)\)[/tex] on the y-axis.
- From the y-intercept, use the slope to find another point. A slope of [tex]\(-3\)[/tex] means that for every 1 unit you move to the right (positive x-direction), you move 3 units down (negative y-direction). So starting from [tex]\((0, -5)\)[/tex], moving to the point [tex]\((1, -8)\)[/tex] (since [tex]\(-5 - 3 = -8\)[/tex]) gives another point on the line.
4. Draw the Line:
Since the inequality is strict ([tex]\(>\)[/tex]), we use a dashed line to represent the boundary where [tex]\( y = -3x - 5 \)[/tex]. A dashed line indicates that points on the line are not included in the solution set.
5. Shade the Appropriate Region:
The inequality is [tex]\( y > -3x - 5 \)[/tex]. This means that we shade the region above the line. Any point in this region will have a y-coordinate greater than [tex]\(-3x - 5\)[/tex].
6. Conclusion:
The graph should show:
- A dashed line representing [tex]\( y = -3x - 5 \)[/tex].
- The area above this line should be shaded.
So, the characteristics of the graph we derived are:
- The slope of the line is [tex]\(-3\)[/tex].
- The y-intercept of the line is [tex]\(-5\)[/tex].
- The inequality is strict ([tex]\(>\)[/tex]), meaning the line is dashed.
- The shaded region is above the line.
These characteristics match the results:
[tex]\[ (-3, -5, '>', 'dashed') \][/tex]
This detailed explanation should help you correctly draw and understand the graph of the inequality [tex]\( y + 2 > -3x - 3 \)[/tex].
1. Rewrite the Inequality:
To put the equation in a more familiar form, we need to isolate [tex]\( y \)[/tex].
[tex]\[ y + 2 > -3x - 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ y > -3x - 3 - 2 \][/tex]
Simplify the right-hand side:
[tex]\[ y > -3x - 5 \][/tex]
2. Identify the Line:
The inequality [tex]\( y > -3x - 5 \)[/tex] involves a linear expression on the right-hand side. To graph this inequality, first, consider the related equation where the inequality is replaced by an equality:
[tex]\[ y = -3x - 5 \][/tex]
3. Graph the Line [tex]\( y = -3x - 5 \)[/tex]:
- This line has a slope of [tex]\(-3\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- To graph it, start at [tex]\((0, -5)\)[/tex] on the y-axis.
- From the y-intercept, use the slope to find another point. A slope of [tex]\(-3\)[/tex] means that for every 1 unit you move to the right (positive x-direction), you move 3 units down (negative y-direction). So starting from [tex]\((0, -5)\)[/tex], moving to the point [tex]\((1, -8)\)[/tex] (since [tex]\(-5 - 3 = -8\)[/tex]) gives another point on the line.
4. Draw the Line:
Since the inequality is strict ([tex]\(>\)[/tex]), we use a dashed line to represent the boundary where [tex]\( y = -3x - 5 \)[/tex]. A dashed line indicates that points on the line are not included in the solution set.
5. Shade the Appropriate Region:
The inequality is [tex]\( y > -3x - 5 \)[/tex]. This means that we shade the region above the line. Any point in this region will have a y-coordinate greater than [tex]\(-3x - 5\)[/tex].
6. Conclusion:
The graph should show:
- A dashed line representing [tex]\( y = -3x - 5 \)[/tex].
- The area above this line should be shaded.
So, the characteristics of the graph we derived are:
- The slope of the line is [tex]\(-3\)[/tex].
- The y-intercept of the line is [tex]\(-5\)[/tex].
- The inequality is strict ([tex]\(>\)[/tex]), meaning the line is dashed.
- The shaded region is above the line.
These characteristics match the results:
[tex]\[ (-3, -5, '>', 'dashed') \][/tex]
This detailed explanation should help you correctly draw and understand the graph of the inequality [tex]\( y + 2 > -3x - 3 \)[/tex].
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
