Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Absolutely, let’s break this down step by step:
### Step 1: Rewrite the Inequality
First, we start with the given linear inequality:
[tex]\[ y + 2 \leq \frac{1}{4}x - 1 \][/tex]
To simplify this and get it into the slope-intercept form [tex]\( y = mx + b \)[/tex], we subtract 2 from both sides:
[tex]\[ y \leq \frac{1}{4}x - 1 - 2 \][/tex]
Simplifying further:
[tex]\[ y \leq \frac{1}{4}x - 3 \][/tex]
### Step 2: Identify the Slope and Y-Intercept
From the inequality in slope-intercept form [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex]:
- Slope (m): [tex]\( \frac{1}{4} \)[/tex]
- Y-intercept (b): [tex]\( -3 \)[/tex]
### Step 3: Graph the Line
Now let's graph the line [tex]\( y = \frac{1}{4}x - 3 \)[/tex]:
1. Y-Intercept: Start by plotting the y-intercept at [tex]\( (0, -3) \)[/tex] on the graph.
2. Slope: The slope [tex]\( \frac{1}{4} \)[/tex] means that for every 4 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction). From the y-intercept point (0, -3):
- Move right 4 units to [tex]\( x = 4 \)[/tex]
- Move up 1 unit to [tex]\( y = -2 \)[/tex]
Plot the second point at [tex]\( (4, -2) \)[/tex].
3. Draw the Line: Draw a straight line through these two points. Since the inequality is [tex]\( \leq \)[/tex] (less than or equal), the line will be solid, indicating that points on the line satisfy the inequality.
### Step 4: Shade the Appropriate Region
The inequality is [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex], which means we shade the area below the line, including the line itself (solid line):
- Below the Line: This is all the points that lie below (or on) the line [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
### Step 5: Match the Graph to the Answer Choices
Now, let's match the descriptions to the graphs provided in the answer choices:
- Graph A: Verify if the y-intercept is at -3, the slope is [tex]\( \frac{1}{4} \)[/tex], and the region below the line is shaded.
- Graph B: Verify the same criteria (y-intercept at -3, slope of [tex]\( \frac{1}{4} \)[/tex], region below the line shaded).
- Graph C: Verify the criteria.
- Graph D: Verify the criteria.
### Conclusion
By identifying the correct graph that matches these criteria, we find that:
- Y-intercept: -3
- Slope: [tex]\(\frac{1}{4}\)[/tex]
- Shaded Region: Below the line [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex]
The correct graph (as per the given description) that satisfies these conditions is Graph B.
So, the answer is:
[tex]\[ \boxed{B} \][/tex]
### Step 1: Rewrite the Inequality
First, we start with the given linear inequality:
[tex]\[ y + 2 \leq \frac{1}{4}x - 1 \][/tex]
To simplify this and get it into the slope-intercept form [tex]\( y = mx + b \)[/tex], we subtract 2 from both sides:
[tex]\[ y \leq \frac{1}{4}x - 1 - 2 \][/tex]
Simplifying further:
[tex]\[ y \leq \frac{1}{4}x - 3 \][/tex]
### Step 2: Identify the Slope and Y-Intercept
From the inequality in slope-intercept form [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex]:
- Slope (m): [tex]\( \frac{1}{4} \)[/tex]
- Y-intercept (b): [tex]\( -3 \)[/tex]
### Step 3: Graph the Line
Now let's graph the line [tex]\( y = \frac{1}{4}x - 3 \)[/tex]:
1. Y-Intercept: Start by plotting the y-intercept at [tex]\( (0, -3) \)[/tex] on the graph.
2. Slope: The slope [tex]\( \frac{1}{4} \)[/tex] means that for every 4 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction). From the y-intercept point (0, -3):
- Move right 4 units to [tex]\( x = 4 \)[/tex]
- Move up 1 unit to [tex]\( y = -2 \)[/tex]
Plot the second point at [tex]\( (4, -2) \)[/tex].
3. Draw the Line: Draw a straight line through these two points. Since the inequality is [tex]\( \leq \)[/tex] (less than or equal), the line will be solid, indicating that points on the line satisfy the inequality.
### Step 4: Shade the Appropriate Region
The inequality is [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex], which means we shade the area below the line, including the line itself (solid line):
- Below the Line: This is all the points that lie below (or on) the line [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
### Step 5: Match the Graph to the Answer Choices
Now, let's match the descriptions to the graphs provided in the answer choices:
- Graph A: Verify if the y-intercept is at -3, the slope is [tex]\( \frac{1}{4} \)[/tex], and the region below the line is shaded.
- Graph B: Verify the same criteria (y-intercept at -3, slope of [tex]\( \frac{1}{4} \)[/tex], region below the line shaded).
- Graph C: Verify the criteria.
- Graph D: Verify the criteria.
### Conclusion
By identifying the correct graph that matches these criteria, we find that:
- Y-intercept: -3
- Slope: [tex]\(\frac{1}{4}\)[/tex]
- Shaded Region: Below the line [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex]
The correct graph (as per the given description) that satisfies these conditions is Graph B.
So, the answer is:
[tex]\[ \boxed{B} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
