Get detailed and accurate responses to your questions on IDNLearn.com. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
Sagot :
To graph the inequality [tex]\( y > -2x - 1 \)[/tex], follow these steps:
### Step 1: Graph the boundary line
First, we need to graph the boundary line given by the equation [tex]\( y = -2x - 1 \)[/tex].
1. Find the y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = -2x - 1 \)[/tex]:
[tex]\[ y = -2(0) - 1 = -1 \][/tex]
So, the y-intercept is at the point [tex]\((0, -1)\)[/tex].
2. Find the x-intercept: Set [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -2x - 1 \)[/tex]:
[tex]\[ 0 = -2x - 1 \Rightarrow 2x = -1 \Rightarrow x = -\frac{1}{2} \][/tex]
So, the x-intercept is at the point [tex]\((-\frac{1}{2}, 0)\)[/tex].
3. Plot the boundary line: Draw a straight line through the points [tex]\((0, -1)\)[/tex] and [tex]\((-\frac{1}{2}, 0)\)[/tex]. This line represents the equation [tex]\( y = -2x - 1 \)[/tex].
Since the inequality is strict ([tex]\( y > -2x - 1 \)[/tex]), we should draw the boundary line as a dashed line to indicate that points on the line itself are not included in the solution set.
### Step 2: Determine which side of the boundary line to shade
We need to determine which side of the boundary line represents the solution to the inequality [tex]\( y > -2x - 1 \)[/tex]. To do this, we can pick a test point that is not on the boundary line and see if it satisfies the inequality.
A common and convenient test point is [tex]\((0, 0)\)[/tex]:
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\( y > -2x - 1 \)[/tex]:
[tex]\[ 0 > -2(0) - 1 \Rightarrow 0 > -1 \][/tex]
- Since [tex]\( 0 > -1 \)[/tex] is true, the point [tex]\((0, 0)\)[/tex] satisfies the inequality. This means the region containing the point [tex]\((0, 0)\)[/tex] is part of the solution set.
### Step 3: Shade the solution region
- Shade the region above the dashed line [tex]\( y = -2x - 1 \)[/tex]. This region represents all points [tex]\((x, y)\)[/tex] where [tex]\( y > -2x - 1 \)[/tex].
### Step 4: Add labels and final touches
- Label the axes with [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Optionally, include the boundary line equation [tex]\( y = -2x - 1 \)[/tex] on the graph for clarity.
- Ensure the graph is neatly presented with gridlines if necessary, to improve readability.
Here is a summary visual of the graphing procedure:
1. Plot the points and draw a dashed line for [tex]\( y = -2x - 1 \)[/tex].
2. Shade the region above this line to represent [tex]\( y > -2x - 1 \)[/tex].
By following these steps, you can clearly visualize the solution set for the inequality [tex]\( y > -2x - 1 \)[/tex].
### Step 1: Graph the boundary line
First, we need to graph the boundary line given by the equation [tex]\( y = -2x - 1 \)[/tex].
1. Find the y-intercept: Set [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = -2x - 1 \)[/tex]:
[tex]\[ y = -2(0) - 1 = -1 \][/tex]
So, the y-intercept is at the point [tex]\((0, -1)\)[/tex].
2. Find the x-intercept: Set [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = -2x - 1 \)[/tex]:
[tex]\[ 0 = -2x - 1 \Rightarrow 2x = -1 \Rightarrow x = -\frac{1}{2} \][/tex]
So, the x-intercept is at the point [tex]\((-\frac{1}{2}, 0)\)[/tex].
3. Plot the boundary line: Draw a straight line through the points [tex]\((0, -1)\)[/tex] and [tex]\((-\frac{1}{2}, 0)\)[/tex]. This line represents the equation [tex]\( y = -2x - 1 \)[/tex].
Since the inequality is strict ([tex]\( y > -2x - 1 \)[/tex]), we should draw the boundary line as a dashed line to indicate that points on the line itself are not included in the solution set.
### Step 2: Determine which side of the boundary line to shade
We need to determine which side of the boundary line represents the solution to the inequality [tex]\( y > -2x - 1 \)[/tex]. To do this, we can pick a test point that is not on the boundary line and see if it satisfies the inequality.
A common and convenient test point is [tex]\((0, 0)\)[/tex]:
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\( y > -2x - 1 \)[/tex]:
[tex]\[ 0 > -2(0) - 1 \Rightarrow 0 > -1 \][/tex]
- Since [tex]\( 0 > -1 \)[/tex] is true, the point [tex]\((0, 0)\)[/tex] satisfies the inequality. This means the region containing the point [tex]\((0, 0)\)[/tex] is part of the solution set.
### Step 3: Shade the solution region
- Shade the region above the dashed line [tex]\( y = -2x - 1 \)[/tex]. This region represents all points [tex]\((x, y)\)[/tex] where [tex]\( y > -2x - 1 \)[/tex].
### Step 4: Add labels and final touches
- Label the axes with [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Optionally, include the boundary line equation [tex]\( y = -2x - 1 \)[/tex] on the graph for clarity.
- Ensure the graph is neatly presented with gridlines if necessary, to improve readability.
Here is a summary visual of the graphing procedure:
1. Plot the points and draw a dashed line for [tex]\( y = -2x - 1 \)[/tex].
2. Shade the region above this line to represent [tex]\( y > -2x - 1 \)[/tex].
By following these steps, you can clearly visualize the solution set for the inequality [tex]\( y > -2x - 1 \)[/tex].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.