Answer :
To graph the inequality [tex]\( y + 2 \leq \frac{1}{4}x - 1 \)[/tex] and identify the correct graph, follow these steps:
### Step-by-Step Solution
1. Rewrite the Inequality:
Start by isolating [tex]\( y \)[/tex] on one side of the inequality.
[tex]\[ y + 2 \leq \frac{1}{4}x - 1 \][/tex]
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y \leq \frac{1}{4}x - 1 - 2 \][/tex]
Simplify the right side:
[tex]\[ y \leq \frac{1}{4}x - 3 \][/tex]
2. Identify the Slope and Y-intercept:
From the inequality [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex], we can directly identify the slope and y-intercept of the corresponding line (ignoring the inequality for a moment):
- Slope (m): [tex]\(\frac{1}{4}\)[/tex]
- Y-intercept (b): [tex]\(-3\)[/tex]
3. Graph the Line:
- First, plot the y-intercept [tex]\((0, -3)\)[/tex] on the graph.
- Next, use the slope to find another point on the line. The slope [tex]\(\frac{1}{4}\)[/tex] indicates that for every 4 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).
Starting from the y-intercept [tex]\((0, -3)\)[/tex]:
- Move 4 units to the right: [tex]\( x = 4 \)[/tex]
- Move 1 unit up: [tex]\( y = -3 + 1 = -2 \)[/tex]
This gives us the point [tex]\((4, -2)\)[/tex]. Plot this point on the graph as well.
- Draw a straight line through these two points extending in both directions. Since the inequality is [tex]\( \leq \)[/tex], this line should be solid, indicating that points on the line itself satisfy the inequality.
4. Shade the Appropriate Region:
The inequality [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex] indicates that we need to shade the region below the line, because [tex]\( y \)[/tex] is less than or equal to the expression on the right-hand side.
To determine whether to shade above or below the line, you can test a point that is not on the line. A common choice is the origin [tex]\((0, 0)\)[/tex]:
Substitute [tex]\((0, 0)\)[/tex] into the inequality:
[tex]\[ 0 \leq \frac{1}{4}(0) - 3 \][/tex]
Simplifies to:
[tex]\[ 0 \leq -3 \][/tex]
This statement is false, meaning the origin is not in the solution region. Therefore, shade the region below the line.
5. Match with Answer Choices:
Compare the graph you drew with the given answer choices. You should look for a graph with:
- A solid line going through points (0, -3) and (4, -2).
- The region below this line shaded.
Based on these criteria, you should identify the correct graphical representation among the given choices.
### Step-by-Step Solution
1. Rewrite the Inequality:
Start by isolating [tex]\( y \)[/tex] on one side of the inequality.
[tex]\[ y + 2 \leq \frac{1}{4}x - 1 \][/tex]
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y \leq \frac{1}{4}x - 1 - 2 \][/tex]
Simplify the right side:
[tex]\[ y \leq \frac{1}{4}x - 3 \][/tex]
2. Identify the Slope and Y-intercept:
From the inequality [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex], we can directly identify the slope and y-intercept of the corresponding line (ignoring the inequality for a moment):
- Slope (m): [tex]\(\frac{1}{4}\)[/tex]
- Y-intercept (b): [tex]\(-3\)[/tex]
3. Graph the Line:
- First, plot the y-intercept [tex]\((0, -3)\)[/tex] on the graph.
- Next, use the slope to find another point on the line. The slope [tex]\(\frac{1}{4}\)[/tex] indicates that for every 4 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).
Starting from the y-intercept [tex]\((0, -3)\)[/tex]:
- Move 4 units to the right: [tex]\( x = 4 \)[/tex]
- Move 1 unit up: [tex]\( y = -3 + 1 = -2 \)[/tex]
This gives us the point [tex]\((4, -2)\)[/tex]. Plot this point on the graph as well.
- Draw a straight line through these two points extending in both directions. Since the inequality is [tex]\( \leq \)[/tex], this line should be solid, indicating that points on the line itself satisfy the inequality.
4. Shade the Appropriate Region:
The inequality [tex]\( y \leq \frac{1}{4}x - 3 \)[/tex] indicates that we need to shade the region below the line, because [tex]\( y \)[/tex] is less than or equal to the expression on the right-hand side.
To determine whether to shade above or below the line, you can test a point that is not on the line. A common choice is the origin [tex]\((0, 0)\)[/tex]:
Substitute [tex]\((0, 0)\)[/tex] into the inequality:
[tex]\[ 0 \leq \frac{1}{4}(0) - 3 \][/tex]
Simplifies to:
[tex]\[ 0 \leq -3 \][/tex]
This statement is false, meaning the origin is not in the solution region. Therefore, shade the region below the line.
5. Match with Answer Choices:
Compare the graph you drew with the given answer choices. You should look for a graph with:
- A solid line going through points (0, -3) and (4, -2).
- The region below this line shaded.
Based on these criteria, you should identify the correct graphical representation among the given choices.
