Answer :
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]
