To solve the linear inequality [tex]\( y > \frac{1}{3}x - 2 \)[/tex], let's follow a step-by-step approach to understand its solution:
1. Identify the Inequality:
The inequality given is [tex]\( y > \frac{1}{3}x - 2 \)[/tex].
2. Understand the Graphical Representation:
This is a linear inequality, which means we are dealing with the region above or below a line. The line itself in this case has the equation:
[tex]\[
y = \frac{1}{3}x - 2
\][/tex]
This line has a slope of [tex]\( \frac{1}{3} \)[/tex] and a y-intercept of [tex]\( -2 \)[/tex].
3. Graph the Boundary Line:
To graph the boundary line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]:
- Start by plotting the y-intercept (0, -2).
- Use the slope to find another point on the line. From (0, -2), move up 1 unit and right 3 units to get to (3, -1).
4. Solid or Dashed Line?:
Since the inequality is strictly greater than ( [tex]\( > \)[/tex] ), we use a dashed line for the graph. This indicates that points on the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex] are not included in the solution.
5. Determine the Shading:
For [tex]\( y > \frac{1}{3}x - 2 \)[/tex], we shade the region above the dashed line because we are looking for the set of points where [tex]\( y \)[/tex] is greater than [tex]\( \frac{1}{3}x - 2 \)[/tex].
Summary:
- A dashed line represents the boundary [tex]\( y = \frac{1}{3}x - 2 \)[/tex].
- Shade the region above this line.
Given our steps and conclusion, the correct solution to the inequality [tex]\( y > \frac{1}{3}x - 2 \)[/tex] involves graphing a dashed line for the boundary [tex]\( y = \frac{1}{3}x - 2 \)[/tex] and shading the area above this line.
If options A and B had information on the graphical representation or coordinate points, you would now match these characteristics with the correct choice. However, without details on options A and B, ensure that you understand this methodology to identify the correct graphical representation in any similar situation.
