- Determine the boundary line equation \(-6x - 3y = -18\).
- Find the intercepts: when \(x=0\), \(y=6\); when \(y=0\), \(x=3\).
- Test the point \((0,0)\) to find that the origin is not in the solution set.
- The solution set is the region not containing the origin, bounded by the solid line passing through \((0,6)\) and \((3,0)\).
- Final intercepts are: \(y=6\), \(x=3\).
Explanation
- Identifying the Boundary Line
To graph the solution set for the inequality \(-6x - 3y \leq -18\), we first need to determine the boundary line defined by the equation \(-6x - 3y = -18\). This line will divide the coordinate plane into two half-planes. Since the inequality includes the "or equal to" sign (\(\leq\)), the points on this boundary line are part of the solution set, so we will draw it as a solid line.
- Finding the Intercepts
To find the intercepts of the boundary line \(-6x - 3y = -18\), we substitute zero for one variable and solve for the other.
- For the \(y\)-intercept, we set \(x = 0\):
\[-6(0) - 3y = -18 \implies -3y = -18 \implies y = 6.\]
So, the \(y\)-intercept is at the point \((0, 6)\).
- For the \(x\)-intercept, we set \(y = 0\):
\[-6x - 3(0) = -18 \implies -6x = -18 \implies x = 3.\]
So, the \(x\)-intercept is at the point \((3, 0)\).
- Determining the Solution Region
Now that we have the intercepts \((0, 6)\) and \((3, 0)\), we draw a solid line passing through these two points. To determine which side of the line represents the solution set, we pick a test point that is not on the line, such as \((0, 0)\). Substituting this into the inequality:
\[-6(0) - 3(0) \leq -18 \implies 0 \leq -18.\]
Since \(0 \leq -18\) is a false statement, the region containing the origin \((0, 0)\) is not part of the solution set. Therefore, the solution set is the half-plane on the opposite side of the line that does not contain the origin. We shade this region to represent all points \((x, y)\) satisfying the inequality.
- Final Conclusion
The boundary line passes through the points \((0, 6)\) and \((3, 0)\). The solution set consists of all points \((x, y)\) in the coordinate plane that lie on the side of this solid line furthest from the origin. The \(x\)-intercept is 3 and the \(y\)-intercept is 6.
Examples
Imagine you are managing the budget for a school event. You have two types of snacks to purchase: snack A (x) and snack B (y). Due to dietary and logistical constraints, the cost equation is represented by \(-6x - 3y \leq -18\), which relates the quantity of each snack you can buy based on available funds. By graphing this inequality, you can visually determine the combinations of snacks (represented by the shaded region) that fit within your budget and constraints, helping you make an informed decision on your purchases.