Answer :
To graph the inequality [tex]\( y < -\frac{3}{4}x + 2 \)[/tex], follow these steps:
1. Graph the boundary line [tex]\( y = -\frac{3}{4}x + 2 \)[/tex]:
- This is a linear equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Identify the y-intercept ([tex]\(b\)[/tex]), which is [tex]\(2\)[/tex]. This means the line crosses the y-axis at [tex]\( y = 2 \)[/tex].
- Determine the slope ([tex]\(m\)[/tex]), which is [tex]\(-\frac{3}{4}\)[/tex]. This indicates that for every 4 units you move to the right along the x-axis, you move 3 units down along the y-axis.
2. Plot the y-intercept at [tex]\( (0, 2) \)[/tex].
3. Use the slope to find another point:
- From the y-intercept [tex]\( (0, 2) \)[/tex]:
- Move 4 units to the right (positive direction) along the x-axis to [tex]\( x = 4 \)[/tex].
- Move 3 units down (negative direction) along the y-axis to [tex]\( y = -1 \)[/tex].
- This gives the point [tex]\( (4, -1) \)[/tex].
- Plot this point on your graph.
4. Draw the boundary line through these two points [tex]\( (0, 2) \)[/tex] and [tex]\( (4, -1) \)[/tex].
5. Determine the type of line:
- Since the inequality is [tex]\( y < -\frac{3}{4}x + 2 \)[/tex] (strictly less than), draw a dashed line to indicate that points on the line itself are not included in the solution set.
6. Shade the region for the inequality:
- The inequality [tex]\( y < -\frac{3}{4}x + 2 \)[/tex] indicates the region below the line.
- Choose a test point not on the line, such as [tex]\( (0, 0) \)[/tex]:
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality: [tex]\( 0 < -\frac{3}{4}(0) + 2 \)[/tex] simplifies to [tex]\( 0 < 2 \)[/tex], which is true.
- Since [tex]\( (0, 0) \)[/tex] satisfies the inequality, shade the region below the dashed line.
After these steps, check which graph among the answer choices shows a dashed line with slope [tex]\(-\frac{3}{4}\)[/tex], y-intercept 2, and the shaded region below the line. Compare the graph you drew with each answer choice:
- Answer Choice A: Examine if it has a dashed line with the correct slope and intercept with shading below.
- Answer Choice B: Similarly, check for the correct dashed line features and shading.
- Answer Choice C: Verify the line type, slope, intercept, and shaded region.
- Answer Choice D: Check the same characteristics as above.
Based on the correct graph details, identify the answer choice that matches your graph.
1. Graph the boundary line [tex]\( y = -\frac{3}{4}x + 2 \)[/tex]:
- This is a linear equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Identify the y-intercept ([tex]\(b\)[/tex]), which is [tex]\(2\)[/tex]. This means the line crosses the y-axis at [tex]\( y = 2 \)[/tex].
- Determine the slope ([tex]\(m\)[/tex]), which is [tex]\(-\frac{3}{4}\)[/tex]. This indicates that for every 4 units you move to the right along the x-axis, you move 3 units down along the y-axis.
2. Plot the y-intercept at [tex]\( (0, 2) \)[/tex].
3. Use the slope to find another point:
- From the y-intercept [tex]\( (0, 2) \)[/tex]:
- Move 4 units to the right (positive direction) along the x-axis to [tex]\( x = 4 \)[/tex].
- Move 3 units down (negative direction) along the y-axis to [tex]\( y = -1 \)[/tex].
- This gives the point [tex]\( (4, -1) \)[/tex].
- Plot this point on your graph.
4. Draw the boundary line through these two points [tex]\( (0, 2) \)[/tex] and [tex]\( (4, -1) \)[/tex].
5. Determine the type of line:
- Since the inequality is [tex]\( y < -\frac{3}{4}x + 2 \)[/tex] (strictly less than), draw a dashed line to indicate that points on the line itself are not included in the solution set.
6. Shade the region for the inequality:
- The inequality [tex]\( y < -\frac{3}{4}x + 2 \)[/tex] indicates the region below the line.
- Choose a test point not on the line, such as [tex]\( (0, 0) \)[/tex]:
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality: [tex]\( 0 < -\frac{3}{4}(0) + 2 \)[/tex] simplifies to [tex]\( 0 < 2 \)[/tex], which is true.
- Since [tex]\( (0, 0) \)[/tex] satisfies the inequality, shade the region below the dashed line.
After these steps, check which graph among the answer choices shows a dashed line with slope [tex]\(-\frac{3}{4}\)[/tex], y-intercept 2, and the shaded region below the line. Compare the graph you drew with each answer choice:
- Answer Choice A: Examine if it has a dashed line with the correct slope and intercept with shading below.
- Answer Choice B: Similarly, check for the correct dashed line features and shading.
- Answer Choice C: Verify the line type, slope, intercept, and shaded region.
- Answer Choice D: Check the same characteristics as above.
Based on the correct graph details, identify the answer choice that matches your graph.
