Sure, let's go through the solution step by step.
Given the inequality:
[tex]\[ -6x - 3y \leq -18 \][/tex]
Step 1: Identify the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts of the boundary line. To find the intercepts, we need to treat the inequality as an equation and solve for the intercepts.
Finding the y-intercept:
To find the [tex]\( y \)[/tex]-intercept, set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ -6(0) - 3y = -18 \][/tex]
[tex]\[ -3y = -18 \][/tex]
[tex]\[ y = 6 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 6) \)[/tex].
Finding the x-intercept:
To find the [tex]\( x \)[/tex]-intercept, set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ -6x - 3(0) = -18 \][/tex]
[tex]\[ -6x = -18 \][/tex]
[tex]\[ x = 3 \][/tex]
So, the [tex]\( x \)[/tex]-intercept is at [tex]\( (3, 0) \)[/tex].
Now we can represent these points in a table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 6 \\
3 & 0 \\
\hline
\end{array}
\][/tex]
This completes the first step. Next, we would graph the boundary line using the intercepts and shade the appropriate region based on the inequality. But for now, we have correctly identified the intercepts as:
When [tex]\( x = 0 \)[/tex], [tex]\( y = 6 \)[/tex].
When [tex]\( y = 0 \)[/tex], [tex]\( x = 3 \)[/tex].
