Answer :
To graph the linear inequality [tex]\( 6x + 2y > -10 \)[/tex], we will follow these steps:
### Step 1: Rewrite the Inequality in Slope-Intercept Form
The inequality [tex]\( 6x + 2y > -10 \)[/tex] can be rewritten in slope-intercept form [tex]\( y = mx + b \)[/tex] by isolating [tex]\( y \)[/tex]:
1. Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ 2y > -6x - 10 \][/tex]
2. Divide every term by 2:
[tex]\[ y > -3x - 5 \][/tex]
### Step 2: Graph the Boundary Line
The boundary of the inequality [tex]\( y > -3x - 5 \)[/tex] is the line [tex]\( y = -3x - 5 \)[/tex]. This line is not included in the solution set because the inequality is strict (i.e., [tex]\( > \)[/tex] and not [tex]\( \geq \)[/tex]). Therefore, we should draw this line as a dashed line.
1. Identify the y-intercept ([tex]\( b \)[/tex]):
[tex]\[ y = -3x - 5 \implies b = -5 \][/tex]
2. Identify the slope ([tex]\( m \)[/tex]):
[tex]\[ y = -3x - 5 \implies m = -3 \][/tex]
3. Plot the y-intercept ([tex]\( 0, -5 \)[/tex]).
4. Use the slope to plot another point. The slope of -3 means you go down 3 units for every 1 unit you go to the right. Starting from [tex]\((0, -5)\)[/tex]:
- Move 1 unit to the right to [tex]\((1, -5)\)[/tex].
- Move down 3 units to [tex]\((1, -8)\)[/tex].
5. Draw a dashed line through these points to represent the boundary line.
### Step 3: Shade the Appropriate Region
Since the inequality is [tex]\( y > -3x - 5 \)[/tex]:
1. Identify that the region above the line represents [tex]\( y > -3x - 5 \)[/tex].
2. Shade the entire region above the dashed line, indicating all points in this area satisfy the inequality.
### Summary
The graph of the linear inequality [tex]\( 6x + 2y > -10 \)[/tex] consists of:
- A dashed line for the equation [tex]\( y = -3x - 5 \)[/tex], showing that points on the line are not included in the solution.
- Shading the region above the dashed line, since we are interested in the values [tex]\( y \)[/tex] greater than [tex]\( -3x - 5 \)[/tex].
By following these steps, you can accurately graph the linear inequality on a coordinate plane.
### Step 1: Rewrite the Inequality in Slope-Intercept Form
The inequality [tex]\( 6x + 2y > -10 \)[/tex] can be rewritten in slope-intercept form [tex]\( y = mx + b \)[/tex] by isolating [tex]\( y \)[/tex]:
1. Subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ 2y > -6x - 10 \][/tex]
2. Divide every term by 2:
[tex]\[ y > -3x - 5 \][/tex]
### Step 2: Graph the Boundary Line
The boundary of the inequality [tex]\( y > -3x - 5 \)[/tex] is the line [tex]\( y = -3x - 5 \)[/tex]. This line is not included in the solution set because the inequality is strict (i.e., [tex]\( > \)[/tex] and not [tex]\( \geq \)[/tex]). Therefore, we should draw this line as a dashed line.
1. Identify the y-intercept ([tex]\( b \)[/tex]):
[tex]\[ y = -3x - 5 \implies b = -5 \][/tex]
2. Identify the slope ([tex]\( m \)[/tex]):
[tex]\[ y = -3x - 5 \implies m = -3 \][/tex]
3. Plot the y-intercept ([tex]\( 0, -5 \)[/tex]).
4. Use the slope to plot another point. The slope of -3 means you go down 3 units for every 1 unit you go to the right. Starting from [tex]\((0, -5)\)[/tex]:
- Move 1 unit to the right to [tex]\((1, -5)\)[/tex].
- Move down 3 units to [tex]\((1, -8)\)[/tex].
5. Draw a dashed line through these points to represent the boundary line.
### Step 3: Shade the Appropriate Region
Since the inequality is [tex]\( y > -3x - 5 \)[/tex]:
1. Identify that the region above the line represents [tex]\( y > -3x - 5 \)[/tex].
2. Shade the entire region above the dashed line, indicating all points in this area satisfy the inequality.
### Summary
The graph of the linear inequality [tex]\( 6x + 2y > -10 \)[/tex] consists of:
- A dashed line for the equation [tex]\( y = -3x - 5 \)[/tex], showing that points on the line are not included in the solution.
- Shading the region above the dashed line, since we are interested in the values [tex]\( y \)[/tex] greater than [tex]\( -3x - 5 \)[/tex].
By following these steps, you can accurately graph the linear inequality on a coordinate plane.
