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Graph the solution set for this inequality:

$-6 x-3 y \leq-18$



Step 1: Identify the $x$- and $y$-intercepts of the boundary line.

When $x=0, y=$ □ .

When $y=0, x=$ □ .

Verified Answer :

  • Identify the boundary line \(-6x - 3y = -18\), which simplifies to \(2x + y = 6\).
  • Calculate the intercepts: the \(y\)-intercept is \((0, 6)\) and the \(x\)-intercept is \((3, 0)\).
  • Test the point \((0, 0)\) to find that the inequality \(-6x - 3y \leq -18\) is false, meaning the solution lies on the side of the line not containing the origin.
  • The \(y\)-intercept is \(\boxed{6}\) and the \(x\)-intercept is \(\boxed{3}\).

Explanation

  1. Analyzing the Inequality and Its Boundary

The goal is to determine the solution set for the linear inequality \(-6x - 3y \leq -18\) and to identify its intercepts to aid in graphing. We start by analyzing the boundary of the inequality, which is defined by the equation \(-6x - 3y = -18\). This equation represents a straight line. By simplifying the equation by dividing all terms by \(-3\), we obtain \(2x + y = 6\), which is a much simpler form to analyze. This line will form the boundary of the region representing the inequality.

  1. Calculating the Intercepts

To find the intercepts of the boundary line, we analyze the equation \(2x + y = 6\).

  • To find the \(y\)-intercept, we set \(x = 0\):

\(2(0) + y = 6 \implies y = 6\).
Thus, the \(y\)-intercept is at the point \((0, 6)\).

  • To find the \(x\)-intercept, we set \(y = 0\):

\(2x + 0 = 6 \implies 2x = 6 \implies x = 3\).
Thus, the \(x\)-intercept is at the point \((3, 0)\).

  1. Determining the Solution Region

To determine which side of the line \(2x + y = 6\) forms the solution set for the inequality \(-6x - 3y \leq -18\), we perform a test using the origin \((0, 0)\). Substituting these coordinates into the original inequality gives \(-6(0) - 3(0) \leq -18\), which simplifies to \(0 \leq -18\). This statement is false. Because the point \((0, 0)\) does not satisfy the inequality, the solution region consists of all points on the other side of the boundary line. Therefore, we draw a solid line passing through \((0, 6)\) and \((3, 0)\) and shade the region that does not include the origin.

Examples

Linear inequalities are frequently used in logistics and operations research to define feasible regions. For example, if a warehouse manager has to allocate resources between two different products while staying within a limited budget, the boundary line represents the exact budget limit, and the shaded area represents all possible combinations of products that fit within the financial constraints. Mastering this technique helps in making data-driven decisions where resources are limited, such as calculating production quotas, fuel efficiency, or time management schedules.