To graph the inequality [tex]\( y \geq x - 2 \)[/tex], follow these detailed steps:
1. Graph the boundary line [tex]\( y = x - 2 \)[/tex]:
- This boundary is a straight line.
- Since the inequality is [tex]\( \geq \)[/tex], this line will be solid, indicating that points on the line satisfy the inequality.
2. Find the y-intercept and x-intercept of the line [tex]\( y = x - 2 \)[/tex]:
- Y-intercept: Set [tex]\( x = 0 \)[/tex]. Then,
[tex]\[ y = 0 - 2 \][/tex]
[tex]\[ y = -2 \][/tex]
So, the y-intercept is [tex]\( (0, -2) \)[/tex].
- X-intercept: Set [tex]\( y = 0 \)[/tex]. Then,
[tex]\[ 0 = x - 2 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the x-intercept is [tex]\( (2, 0) \)[/tex].
3. Plot the boundary line using the intercepts:
- Start by plotting the points [tex]\( (0, -2) \)[/tex] and [tex]\( (2, 0) \)[/tex].
- Draw a straight line through these points, extending in both directions. This line represents the equation [tex]\( y = x - 2 \)[/tex].
4. Determine which side of the line to shade for the inequality [tex]\( y \geq x - 2 \)[/tex]:
- Choose a test point not on the line to determine which side to shade. A convenient point to use is [tex]\( (0, 0) \)[/tex].
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality:
[tex]\[ 0 \geq 0 - 2 \][/tex]
[tex]\[ 0 \geq -2 \][/tex]
This inequality is true, so the point [tex]\( (0, 0) \)[/tex] satisfies the inequality. Thus, the area containing [tex]\( (0, 0) \)[/tex] should be shaded.
5. Shade the correct region:
- Since [tex]\( y \geq x - 2 \)[/tex] is satisfied by the test point [tex]\( (0, 0) \)[/tex], shade the region above the boundary line [tex]\( y = x - 2 \)[/tex].
6. Finalize your graph:
- Draw the boundary line [tex]\( y = x - 2 \)[/tex] as a solid line to show that points on the line are included in the inequality.
- Shade the entire region above this line to represent all points where [tex]\( y \geq x - 2 \)[/tex].
### Summary of Steps:
1. Draw a solid line for [tex]\( y = x - 2 \)[/tex].
2. Shade the region above the line to represent [tex]\( y \geq x - 2 \)[/tex].
The resultant graph will have a solid line passing through the points [tex]\( (0, -2) \)[/tex] and [tex]\( (2, 0) \)[/tex], with the area above this line shaded, indicating that all points in this region satisfy the inequality [tex]\( y \geq x - 2 \)[/tex].
