Let's determine how to graph the inequality [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] step-by-step.
1. Understanding the Inequality:
The inequality [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] tells us that we need to identify all points [tex]\((x, y)\)[/tex] where the value of [tex]\( y \)[/tex] is less than the value given by the expression [tex]\( -\frac{1}{3}x + 1 \)[/tex].
2. Start with the Boundary Line:
First, we graph the boundary line given by the equation [tex]\( y = -\frac{1}{3}x + 1 \)[/tex].
- This line is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m = -\frac{1}{3} \)[/tex] and the y-intercept [tex]\( b = 1 \)[/tex].
3. Plotting the Boundary Line:
- Start by plotting the y-intercept (0, 1).
- Use the slope to find another point. The slope [tex]\( -\frac{1}{3} \)[/tex] means that for every 3 units you move to the right (positive direction in x-axis), you move 1 unit down (negative direction in y-axis).
Example points:
- Starting at (0, 1), move right 3 units to (3, 1), then down 1 unit to (3, 0).
- Similarly, starting again at (0, 1), moving left 3 units (to the point (-3, 1)), and then moving up 1 unit (to point (-3, 2)).
Connect these points with a dashed line, as the inequality is strictly [tex]\( < \)[/tex] and does not include the boundary line itself.
4. Choosing the Shaded Region:
- Since the inequality is [tex]\( y < -\frac{1}{3}x + 1 \)[/tex], we shade the region below the boundary line.
- To make sure we shade the correct area, pick a test point not on the line (e.g., the origin (0,0)).
Substitute (0,0) into the inequality:
[tex]\[
0 < -\frac{1}{3}(0) + 1 \implies 0 < 1
\][/tex]
This statement is true, thus the region containing (0,0) is the solution.
5. Graph Description:
The graph will have a dashed line starting at (0, 1), moving downward with a slope of [tex]\(-\frac{1}{3}\)[/tex]. The area below this dashed line will be shaded to indicate all points where [tex]\( y < -\frac{1}{3}x + 1 \)[/tex].
In summary, the correct graph includes:
- A dashed line with the equation [tex]\( y = -\frac{1}{3} x + 1 \)[/tex].
- The region below this dashed line shaded.
