To graph the inequality [tex]\( -x + 3y > 3 \)[/tex], follow these steps:
1. Rewrite the inequality to find the boundary line:
- The boundary line is [tex]\( -x + 3y = 3 \)[/tex].
2. Convert the boundary line to slope-intercept form:
- First, solve for [tex]\( y \)[/tex]:
[tex]\[
-x + 3y = 3
\][/tex]
[tex]\[
3y = x + 3
\][/tex]
[tex]\[
y = \frac{1}{3}x + 1
\][/tex]
3. Plot the boundary line on the coordinate plane:
- To plot the line, you need at least two points. Use these steps:
- Point 1: When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{1}{3}(0) + 1 = 1
\][/tex]
So, the point is [tex]\( (0, 1) \)[/tex].
- Point 2: When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = \frac{1}{3}(3) + 1 = 2
\][/tex]
So, the point is [tex]\( (3, 2) \)[/tex].
- Plot these two points: [tex]\( (0, 1) \)[/tex] and [tex]\( (3, 2) \)[/tex], and draw a solid line through them. This is the boundary line [tex]\( -x + 3y = 3 \)[/tex]. Note that in the inequality [tex]\( -x + 3y > 3 \)[/tex], the boundary line will be drawn as a dashed line to indicate that points on the line are not included in the solution set.
4. Determine which side of the boundary line to shade:
- Pick a test point that is not on the line to determine which region to shade. A common test point is [tex]\( (0, 0) \)[/tex].
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality [tex]\( -x + 3y > 3 \)[/tex]:
[tex]\[
-0 + 3(0) > 3
\][/tex]
[tex]\[
0 > 3
\][/tex]
- This statement is false.
- Since [tex]\( (0, 0) \)[/tex] does not satisfy the inequality, the region that does not include this test point should be shaded. That means you need to shade the region on the opposite side of the line from the origin [tex]\( (0, 0) \)[/tex].
5. Final graph:
- Draw a dashed line through the points [tex]\( (0, 1) \)[/tex] and [tex]\( (3, 2) \)[/tex] to represent the boundary.
- Shade the region above and to the right of this line, as that is the region where the inequality [tex]\( -x + 3y > 3 \)[/tex] holds true.
By following these steps, you will graphically represent the solution to the inequality [tex]\( -x + 3y > 3 \)[/tex].
