Join IDNLearn.com today and start getting the answers you've been searching for. 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]\( 2x - 4y > 6 \)[/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].
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.
