Answer :
Of course! Let's go through the steps to draw the graph of the inequality \( y < \frac{3}{4}x + 2 \) on a piece of paper and determine which answer choice matches the graph.
### Step-by-Step Solution:
1. Understand the inequality:
- We have the inequality \( y < \frac{3}{4}x + 2 \).
- This is a linear inequality, which means we'll start by graphing the line \( y = \frac{3}{4}x + 2 \).
2. Graph the corresponding equation \(y = \frac{3}{4}x + 2\):
- This is a line in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, the slope \( m = \frac{3}{4} \), and the y-intercept \( b = 2 \).
3. Plot the line \(y = \frac{3}{4}x + 2\):
- Start by plotting the y-intercept (0, 2). This is where the line crosses the y-axis.
- Next, use the slope to find another point. The slope \( \frac{3}{4} \) means that for every 4 units you move to the right (positive direction along the x-axis), you move up 3 units.
- From (0, 2), moving 4 units right and 3 units up will lead to the point (4, 5). Plot this point.
- Draw a straight line through these points. Use a dashed line to indicate that it's an inequality (since \( y \) is less than, not less than or equal to, the expression).
4. Shade the appropriate region:
- The inequality \( y < \frac{3}{4}x + 2 \) indicates that we need to shade all the points below the line \( y = \frac{3}{4}x + 2 \).
- Choose a test point not on the line to determine which side to shade. A good choice is the origin (0, 0).
- Plugging the origin into the inequality: \(0 < \frac{3}{4}(0) + 2 \Rightarrow 0 < 2\), which is true.
- Since the origin satisfies the inequality, shade the region that includes the origin.
5. Compare with given answer choices:
- If answer choices A, B, C, and D are graphs, compare them:
- Identify the graphs which have a dashed line through \( y = \frac{3}{4}x + 2 \).
- Ensure the correct region (below the line) is shaded.
By following these steps, you should be able to determine the correct graph that matches the inequality \( y < \frac{3}{4}x + 2 \).
Since we are completely working step-by-step on graph paper without the specific answer choices present here, I can't directly identify A, B, C, or D. However, the correct answer would be a graph featuring a dashed line through the points you calculated and shading below this line.
### Step-by-Step Solution:
1. Understand the inequality:
- We have the inequality \( y < \frac{3}{4}x + 2 \).
- This is a linear inequality, which means we'll start by graphing the line \( y = \frac{3}{4}x + 2 \).
2. Graph the corresponding equation \(y = \frac{3}{4}x + 2\):
- This is a line in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, the slope \( m = \frac{3}{4} \), and the y-intercept \( b = 2 \).
3. Plot the line \(y = \frac{3}{4}x + 2\):
- Start by plotting the y-intercept (0, 2). This is where the line crosses the y-axis.
- Next, use the slope to find another point. The slope \( \frac{3}{4} \) means that for every 4 units you move to the right (positive direction along the x-axis), you move up 3 units.
- From (0, 2), moving 4 units right and 3 units up will lead to the point (4, 5). Plot this point.
- Draw a straight line through these points. Use a dashed line to indicate that it's an inequality (since \( y \) is less than, not less than or equal to, the expression).
4. Shade the appropriate region:
- The inequality \( y < \frac{3}{4}x + 2 \) indicates that we need to shade all the points below the line \( y = \frac{3}{4}x + 2 \).
- Choose a test point not on the line to determine which side to shade. A good choice is the origin (0, 0).
- Plugging the origin into the inequality: \(0 < \frac{3}{4}(0) + 2 \Rightarrow 0 < 2\), which is true.
- Since the origin satisfies the inequality, shade the region that includes the origin.
5. Compare with given answer choices:
- If answer choices A, B, C, and D are graphs, compare them:
- Identify the graphs which have a dashed line through \( y = \frac{3}{4}x + 2 \).
- Ensure the correct region (below the line) is shaded.
By following these steps, you should be able to determine the correct graph that matches the inequality \( y < \frac{3}{4}x + 2 \).
Since we are completely working step-by-step on graph paper without the specific answer choices present here, I can't directly identify A, B, C, or D. However, the correct answer would be a graph featuring a dashed line through the points you calculated and shading below this line.
