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To graph the inequality [tex]\( 2x - 4y > 6 \)[/tex], we need to follow several steps. These steps will include identifying the boundary line and then determining which region satisfies the inequality.
### Step 1: Find the boundary line
The given inequality is [tex]\( 2x - 4y > 6 \)[/tex]. To begin, we need to identify the boundary line by converting the inequality into an equation.
[tex]\[ 2x - 4y = 6 \][/tex]
### Step 2: Rewrite the equation in slope-intercept form
To make it easier to graph, let's rewrite this equation in slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ 2x - 4y = 6 \][/tex]
[tex]\[ -4y = -2x + 6 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the boundary line is [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex].
### Step 3: Plot the boundary line
On a coordinate plane, plot the line [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex]. This involves plotting a few points. For example:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{2}(0) - \frac{3}{2} = -\frac{3}{2} \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \frac{1}{2}(2) - \frac{3}{2} = 1 - \frac{3}{2} = -\frac{1}{2} \][/tex]
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = \frac{1}{2}(4) - \frac{3}{2} = 2 - \frac{3}{2} = \frac{1}{2} \][/tex]
Plot these points on a graph and draw a straight line through them. Since the original inequality is [tex]\( 2x - 4y > 6 \)[/tex], and not [tex]\( 2x - 4y \geq 6 \)[/tex], the line should be dashed to indicate that points on the line are not included in the solution set.
### Step 4: Determine which region to shade
The inequality [tex]\( 2x - 4y > 6 \)[/tex] indicates we need the region where [tex]\( 2x - 4y \)[/tex] is greater than 6. To determine which side of the line to shade, pick a test point that is not on the line. A good test point is usually [tex]\((0, 0)\)[/tex], if it is not on the line.
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\( 2x - 4y > 6 \)[/tex]:
[tex]\[ 2(0) - 4(0) > 6 \][/tex]
[tex]\[ 0 > 6 \][/tex]
This statement is false, so the region that includes the point [tex]\((0, 0)\)[/tex] is not part of the solution.
Since [tex]\((0, 0)\)[/tex] does not satisfy the inequality, we shade the opposite side of the boundary line.
### Final Graph
1. Draw a dashed line for [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex].
2. Shade the region above this line because it represents [tex]\( 2x - 4y > 6 \)[/tex].
This shaded region is the solution to the inequality [tex]\( 2x - 4y > 6 \)[/tex].
### Step 1: Find the boundary line
The given inequality is [tex]\( 2x - 4y > 6 \)[/tex]. To begin, we need to identify the boundary line by converting the inequality into an equation.
[tex]\[ 2x - 4y = 6 \][/tex]
### Step 2: Rewrite the equation in slope-intercept form
To make it easier to graph, let's rewrite this equation in slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ 2x - 4y = 6 \][/tex]
[tex]\[ -4y = -2x + 6 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the boundary line is [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex].
### Step 3: Plot the boundary line
On a coordinate plane, plot the line [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex]. This involves plotting a few points. For example:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{1}{2}(0) - \frac{3}{2} = -\frac{3}{2} \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \frac{1}{2}(2) - \frac{3}{2} = 1 - \frac{3}{2} = -\frac{1}{2} \][/tex]
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ y = \frac{1}{2}(4) - \frac{3}{2} = 2 - \frac{3}{2} = \frac{1}{2} \][/tex]
Plot these points on a graph and draw a straight line through them. Since the original inequality is [tex]\( 2x - 4y > 6 \)[/tex], and not [tex]\( 2x - 4y \geq 6 \)[/tex], the line should be dashed to indicate that points on the line are not included in the solution set.
### Step 4: Determine which region to shade
The inequality [tex]\( 2x - 4y > 6 \)[/tex] indicates we need the region where [tex]\( 2x - 4y \)[/tex] is greater than 6. To determine which side of the line to shade, pick a test point that is not on the line. A good test point is usually [tex]\((0, 0)\)[/tex], if it is not on the line.
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\( 2x - 4y > 6 \)[/tex]:
[tex]\[ 2(0) - 4(0) > 6 \][/tex]
[tex]\[ 0 > 6 \][/tex]
This statement is false, so the region that includes the point [tex]\((0, 0)\)[/tex] is not part of the solution.
Since [tex]\((0, 0)\)[/tex] does not satisfy the inequality, we shade the opposite side of the boundary line.
### Final Graph
1. Draw a dashed line for [tex]\( y = \frac{1}{2}x - \frac{3}{2} \)[/tex].
2. Shade the region above this line because it represents [tex]\( 2x - 4y > 6 \)[/tex].
This shaded region is the solution to the inequality [tex]\( 2x - 4y > 6 \)[/tex].
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