Find solutions to your problems with the help of IDNLearn.com's knowledgeable users. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
Sagot :
Sure, let's go through the steps to graph the inequality [tex]\( y \leq -2x - 1 \)[/tex].
### Step-by-Step Solution
1. Understand the inequality:
- The inequality is given as [tex]\( y \leq -2x - 1 \)[/tex].
- This means we are looking for all the points [tex]\((x, y)\)[/tex] in the plane where the y-coordinate is less than or equal to [tex]\( -2x - 1 \)[/tex].
2. Graph the boundary line:
- First, we graph the boundary line [tex]\( y = -2x - 1 \)[/tex]. This line runs through the plane, and we will use it to help us graph the inequality.
- To graph this, we need at least two points.
3. Find points for the boundary line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) - 1 = -1 \)[/tex]. So, one point is [tex]\((0, -1)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -2(1) - 1 = -3 \)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
4. Plot the boundary line:
- Plot the points [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex] on the graph.
- Draw a straight line through these points. This line represents the equation [tex]\( y = -2x - 1 \)[/tex].
5. Determine the region to shade:
- The inequality is [tex]\( y \leq -2x - 1 \)[/tex]. This means we want the region where y is less than or equal to the boundary line.
- To determine the correct region to shade, choose a test point that is not on the boundary line (the origin [tex]\((0,0)\)[/tex] is often convenient unless it lies on the boundary).
6. Test the point (0,0):
- Substitute [tex]\((0,0)\)[/tex] into the inequality: [tex]\( 0 \leq -2(0) - 1 \)[/tex].
- Simplifies to [tex]\( 0 \leq -1 \)[/tex], which is false.
- Since [tex]\((0,0)\)[/tex] does not satisfy the inequality, the region that does will be on the other side of the line.
7. Shade the correct region:
- Shade the region below the line [tex]\( y = -2x - 1 \)[/tex], as that represents [tex]\( y \leq -2x - 1 \)[/tex].
### Final Graph
- The boundary line [tex]\( y = -2x - 1 \)[/tex] is a straight line with a negative slope.
- The area below this line, including the line itself (since it's [tex]\( \leq \)[/tex]), is shaded.
The graphical representation would look like this:
- A straight line passing through [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex].
- The area below this line shaded to represent all points [tex]\((x, y)\)[/tex] where [tex]\( y \leq -2x - 1 \)[/tex].
### Conclusion
The correct graph for the inequality [tex]\( y \leq -2x - 1 \)[/tex] is a line with a slope of -2, passing through the point [tex]\((0, -1)\)[/tex] and the region below the line shaded.
### Step-by-Step Solution
1. Understand the inequality:
- The inequality is given as [tex]\( y \leq -2x - 1 \)[/tex].
- This means we are looking for all the points [tex]\((x, y)\)[/tex] in the plane where the y-coordinate is less than or equal to [tex]\( -2x - 1 \)[/tex].
2. Graph the boundary line:
- First, we graph the boundary line [tex]\( y = -2x - 1 \)[/tex]. This line runs through the plane, and we will use it to help us graph the inequality.
- To graph this, we need at least two points.
3. Find points for the boundary line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) - 1 = -1 \)[/tex]. So, one point is [tex]\((0, -1)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -2(1) - 1 = -3 \)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
4. Plot the boundary line:
- Plot the points [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex] on the graph.
- Draw a straight line through these points. This line represents the equation [tex]\( y = -2x - 1 \)[/tex].
5. Determine the region to shade:
- The inequality is [tex]\( y \leq -2x - 1 \)[/tex]. This means we want the region where y is less than or equal to the boundary line.
- To determine the correct region to shade, choose a test point that is not on the boundary line (the origin [tex]\((0,0)\)[/tex] is often convenient unless it lies on the boundary).
6. Test the point (0,0):
- Substitute [tex]\((0,0)\)[/tex] into the inequality: [tex]\( 0 \leq -2(0) - 1 \)[/tex].
- Simplifies to [tex]\( 0 \leq -1 \)[/tex], which is false.
- Since [tex]\((0,0)\)[/tex] does not satisfy the inequality, the region that does will be on the other side of the line.
7. Shade the correct region:
- Shade the region below the line [tex]\( y = -2x - 1 \)[/tex], as that represents [tex]\( y \leq -2x - 1 \)[/tex].
### Final Graph
- The boundary line [tex]\( y = -2x - 1 \)[/tex] is a straight line with a negative slope.
- The area below this line, including the line itself (since it's [tex]\( \leq \)[/tex]), is shaded.
The graphical representation would look like this:
- A straight line passing through [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex].
- The area below this line shaded to represent all points [tex]\((x, y)\)[/tex] where [tex]\( y \leq -2x - 1 \)[/tex].
### Conclusion
The correct graph for the inequality [tex]\( y \leq -2x - 1 \)[/tex] is a line with a slope of -2, passing through the point [tex]\((0, -1)\)[/tex] and the region below the line shaded.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
