From simple queries to complex problems, IDNLearn.com provides reliable answers. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
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 being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.