To graph the solution set of the first linear inequality [tex]\( y \leq 6x + 9 \)[/tex], follow these steps:
1. Determine the boundary line:
- The boundary line for the inequality [tex]\( y \leq 6x + 9 \)[/tex] is [tex]\( y = 6x + 9 \)[/tex].
2. Select the type of boundary line:
- Since the inequality is [tex]\( \leq \)[/tex] (less than or equal to), the boundary line will be a solid line.
3. Find two points on the boundary line:
- Choose simple values for [tex]\( x \)[/tex] to find corresponding [tex]\( y \)[/tex] values.
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 6 \cdot 0 + 9 = 9
\][/tex]
So, one point on the boundary line is [tex]\( (0, 9) \)[/tex].
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 6 \cdot 1 + 9 = 15
\][/tex]
So, another point on the boundary line is [tex]\( (1, 15) \)[/tex].
4. Plot the points and draw the boundary line:
- Plot the points [tex]\( (0, 9) \)[/tex] and [tex]\( (1, 15) \)[/tex] on the graph.
- Draw a solid line through these points to represent the boundary line [tex]\( y = 6x + 9 \)[/tex].
5. Shade the region corresponding to the inequality:
- Since the inequality is [tex]\( y \leq 6x + 9 \)[/tex], shade the region below or on the line.
Once you perform these steps, you've successfully graphed the solution set of the first linear inequality [tex]\( y \leq 6x + 9 \)[/tex]. The visualization helps in understanding the feasible region that satisfies this inequality.
