$ax + by < c$

Definition

The expression $ax + by < c$ represents a linear inequality in two variables, where $a$ and $b$ are coefficients, $x$ and $y$ are the variables, and $c$ is a constant. This inequality describes a region on a graph where all the points $(x, y)$ satisfy the condition that the linear combination of the variables is less than the constant. The graph of this inequality consists of a boundary line represented by the equation $ax + by = c$, with the area below this line indicating the solutions to the inequality.

5 Must Know Facts For Your Next Test

1. The inequality $ax + by < c$ shows that all points on one side of the boundary line satisfy the condition, while points on the other side do not.
2. When graphing $ax + by < c$, the boundary line is usually drawn as a dashed line to indicate that points on the line are not included in the solution set.
3. To determine which side of the boundary line to shade, you can test a point (often (0,0) if it's not on the line) to see if it satisfies the inequality.
4. This type of inequality can have multiple solutions, forming a region that can be unbounded or bounded within certain limits.
5. Linear inequalities like $ax + by < c$ are foundational in systems of inequalities, where multiple such inequalities can define complex regions on a graph.

Review Questions

• How does the graph of the linear inequality $ax + by < c$ differ from that of the corresponding linear equation $ax + by = c$?
  • The graph of the linear equation $ax + by = c$ is a straight line that includes all points that satisfy this equality. In contrast, for the inequality $ax + by < c$, this line acts as a boundary, and only points below it are included in the solution set. The boundary line for an inequality is typically dashed to indicate that points on this line do not satisfy the inequality itself.
• Explain how to determine which side of the boundary line to shade when graphing an inequality like $ax + by < c$. What method can be used?
  • To determine which side of the boundary line to shade for an inequality like $ax + by < c$, you can use a test point. A common choice is to use (0, 0) if it is not on the boundary line. Substitute this point into the inequality; if it holds true (e.g., $a(0) + b(0) < c$), then shade the region containing (0, 0). If it does not hold true, shade the opposite side of the boundary line.
• Evaluate how multiple inequalities like $ax + by < c$ can work together to define a solution region in graphing. What considerations must be taken into account?
  • When dealing with multiple inequalities such as $ax + by < c$, each inequality defines its own solution region. To find a common solution region, you must graph each inequality and identify where these regions overlap. It's important to consider both dashed and solid lines based on whether inequalities are strict (< or >) or non-strict (≤ or ≥). The intersection of these shaded areas will give you a feasible region where all inequalities are satisfied simultaneously.

Related terms

Linear Equation: An equation that represents a straight line when graphed, typically in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Boundary Line: The straight line that represents the equality $ax + by = c$ in a linear inequality, separating the solution region from non-solution regions.

Solution Region: The area on a graph that contains all the points $(x, y)$ that satisfy the inequality, which can be above or below the boundary line depending on the inequality sign.