To determine the solution set for the inequality [tex]\(-6x - 3y \leq -18\)[/tex], let's first identify the intercepts of the boundary line.
### Finding the [tex]\(x\)[/tex]-intercept:
The [tex]\(x\)[/tex]-intercept occurs when [tex]\(y = 0\)[/tex]. Substitute [tex]\(y = 0\)[/tex] into the equation:
[tex]\[
-6x - 3(0) = -18
\][/tex]
This simplifies to:
[tex]\[
-6x = -18
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-18}{-6} = 3
\][/tex]
So, the [tex]\(x\)[/tex]-intercept is [tex]\((3, 0)\)[/tex].
### Finding the [tex]\(y\)[/tex]-intercept:
The [tex]\(y\)[/tex]-intercept occurs when [tex]\(x = 0\)[/tex]. Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[
-6(0) - 3y = -18
\][/tex]
This simplifies to:
[tex]\[
-3y = -18
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-18}{-3} = 6
\][/tex]
So, the [tex]\(y\)[/tex]-intercept is [tex]\((0, 6)\)[/tex].
### Step 1 Summary:
When [tex]\(x = 0\)[/tex], [tex]\(y = 6\)[/tex], and when [tex]\(y = 0\)[/tex], [tex]\(x = 3\)[/tex].
This gives us the intercepts:
[tex]\[
\begin{aligned}
& \text{When } x = 0, y = 6 \\
& \text{When } y = 0, x = 3
\end{aligned}
\][/tex]
These intercepts will help us graph the boundary line.
