The expression $ax + by > c$ represents a linear inequality in two variables, where $a$, $b$, and $c$ are constants. This inequality indicates that the values of $x$ and $y$ will create a region on a graph, specifically the area above the line represented by the equation $ax + by = c$. Understanding this term is crucial for identifying feasible regions and solutions in systems of inequalities.
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When graphing $ax + by > c$, you first graph the boundary line $ax + by = c$ as a dashed line to indicate that points on the line are not included in the solution.
The region that satisfies the inequality is determined by testing a point (often the origin) to see if it makes the inequality true; if it does, that region is shaded.
If both $a$ and $b$ are positive, increasing either variable will push points further from the boundary line, creating larger values of $ax + by$.
If either coefficient is zero, such as $a = 0$, the inequality simplifies to $by > c$, which defines a horizontal or vertical line based on the sign of $b$.
The inequality can also be expressed in slope-intercept form, allowing for easy identification of the slope and y-intercept for graphing.
Review Questions
How would you graph the inequality $2x + 3y > 6$, including identifying the boundary line and shading?
To graph the inequality $2x + 3y > 6$, start by graphing the boundary line $2x + 3y = 6$. This line should be dashed because points on it do not satisfy the inequality. To find where to shade, test a point not on the line, like (0,0). Since $2(0) + 3(0) = 0$ is not greater than 6, shade the region above the line instead, which represents all solutions to the inequality.
What effect does changing the coefficients of $x$ and $y$ in the inequality $ax + by > c$ have on its graph?
Changing the coefficients of $x$ and $y$ in the inequality affects both the slope and position of the boundary line. For example, increasing 'a' would make the slope steeper, while decreasing 'b' would lower the y-intercept. Each adjustment alters how steeply or horizontally the boundary line is drawn, subsequently changing which region gets shaded. Thus, understanding these coefficients is crucial for accurately graphing linear inequalities.
Evaluate how changing from an inequality to an equation, such as moving from $ax + by > c$ to $ax + by = c$, impacts understanding of solutions in a graphical context.
Transitioning from an inequality to an equation transforms how solutions are interpreted. While $ax + by = c$ represents just a single boundary line indicating where equality holds, $ax + by > c$ describes a whole region where multiple points can be solutions. This shift emphasizes feasibility; with inequalities, there are infinitely many solutions within a designated area above or below this line. Understanding this distinction highlights not only which points satisfy conditions but also how they relate to each other across a broader spectrum.
Related terms
Linear Equation: An equation that describes a straight line when graphed, typically in the form $y = mx + b$.
Feasible Region: The area on a graph that satisfies all constraints of a system of inequalities.
Boundary Line: The line represented by the equation $ax + by = c$, which separates the graph into different regions based on the inequality.