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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].
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