5.10 Systems of Linear Inequalities in Two Variables
2 min read•june 18, 2024
Linear inequalities in two variables are powerful tools for modeling real-world . They're graphed on a , with shaded regions representing solutions. Systems of these inequalities combine multiple constraints, creating a more complex .
Understanding how to graph and interpret these systems is crucial for solving optimization problems. By identifying variables, formulating inequalities, and visualizing the solution region, we can tackle resource allocation, production planning, and other practical challenges.
Systems of Linear Inequalities in Two Variables
Graphing linear inequality systems
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- in two variables takes the form , , , or , where , , and are real numbers and and are not both zero
- Inequality sign determines which is shaded
- used for strict inequalities or ()
- used for inclusive inequalities or ()
- Inequality sign determines which is shaded
- Graphing a involves:
- Graph each inequality separately on the same coordinate plane
- Solution set is the of all shaded regions satisfying the inequalities
- Solution set of a system of linear inequalities represents all ordered pairs that simultaneously satisfy every inequality in the system
Ordered pairs in inequality systems
- Checking if an satisfies a system of inequalities:
- Substitute and values into each inequality in the system
- If substitution results in true statements for all inequalities, the ordered pair is a solution
- Ordered pair satisfying all inequalities in the system is a solution point contained within the solution set (graphically represented by the intersecting shaded regions)
Real-world applications of inequalities
- Systems of linear inequalities model real-world situations involving constraints or limitations
- Resource allocation problems (budget constraints)
- Production limitations (manufacturing capacity)
- Feasible regions in optimization problems (maximizing profit)
- Modeling a real-world scenario with inequalities:
- Identify variables and define them in the context of the problem
- Formulate inequalities representing constraints or limitations on the variables
- Graph the system of inequalities to visualize the feasible solution region
- Interpret the solution set in terms of the original real-world problem
- problems involve optimizing an subject to constraints modeled by a system of linear inequalities (maximizing profit while satisfying production constraints)
Key components of linear inequalities
- Coordinate plane: The 2D plane where linear inequalities are graphed, consisting of x and y axes
- : The four regions of the coordinate plane, divided by the x and y axes
- : The rate of change in a linear equation, representing the steepness of the line
- : The point where a line crosses the y-axis, often used as a starting point when graphing
- : The property of a solution satisfying all constraints in a system of linear inequalities
Key Terms to Review (27)
<: The symbol '<' represents a mathematical inequality indicating that the value on the left is less than the value on the right. This symbol is used to compare two quantities and helps in formulating linear inequalities, which are essential in both real-world applications and graphical representations of relationships between variables.
>: The symbol '>' represents a greater than relation in mathematics, indicating that one value is larger than another. This concept is fundamental when dealing with inequalities, allowing us to express and analyze relationships between numbers or expressions. Understanding this symbol is essential for solving linear inequalities and systems of inequalities, as well as for evaluating fairness in voting methods where comparisons are necessary.
≤: The symbol '≤' represents a relationship in mathematics indicating that one value is less than or equal to another. This symbol is crucial for expressing inequalities and understanding the boundaries of solutions in various mathematical contexts, helping to determine feasible solutions, especially in optimization problems.
≥: The symbol '≥' represents the concept of 'greater than or equal to' in mathematics, establishing a relationship between two values. This symbol is crucial in expressing linear inequalities, where one side of the inequality can either exceed or be equal to the other side. It helps define boundaries in mathematical expressions and is foundational in various applications like optimization and systems of inequalities.
Boundary line: A boundary line is a line that separates the regions of solutions for a system of linear inequalities in two variables. It is derived from the corresponding equation of the inequality, and it plays a crucial role in determining the areas where the inequalities hold true. The boundary line can either be solid or dashed, indicating whether points on the line are included in the solution set or not.
Constraint: A constraint is a limitation or restriction that defines the boundaries within which a solution must be found. In mathematical contexts, constraints help to narrow down possible solutions by establishing conditions that must be satisfied, such as inequalities or specific values. They are crucial for solving problems, as they guide the decision-making process in various applications.
Constraints: Constraints are conditions or limits imposed on variables in a mathematical model. They restrict the feasible region within which an optimal solution can be found.
Coordinate plane: A coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis), allowing for the precise representation of points using ordered pairs. Each point on the coordinate plane is identified by its coordinates, which denote its position relative to the axes, facilitating the visualization and analysis of mathematical relationships and functions.
Dashed boundary line: A dashed boundary line is used in graphing systems of linear inequalities to indicate that points on the line itself are not included in the solution set. This means that the line represents an inequality that is strict, such as 'greater than' or 'less than', which does not allow for equality. Understanding this concept is crucial when determining feasible regions in two-dimensional space, as it visually distinguishes between points that satisfy the inequality and those that do not.
Feasibility: Feasibility refers to the condition of being possible or practical within a given context, especially concerning the constraints outlined by linear inequalities. In scenarios involving two-variable inequalities, feasibility indicates whether a certain set of values satisfies all the conditions imposed by those inequalities. This concept is crucial for determining valid solutions in optimization problems where certain criteria must be met.
Feasible region: The feasible region is the set of all possible solutions that satisfy a given set of constraints in a mathematical context. This concept is crucial when dealing with inequalities and optimization problems, as it visually represents the area where all constraints overlap. The feasible region is often graphed in two-dimensional space, showing the combinations of variables that meet all specified conditions.
Graphing method: The graphing method is a visual technique used to solve systems of equations or inequalities by plotting their graphs on a coordinate plane. This method allows for the identification of solutions at the points where the graphs intersect, which represent the values that satisfy all equations or inequalities in the system. It is particularly effective for illustrating the relationships between variables and understanding how changes in one variable affect another.
Half-plane: A half-plane is a geometric concept that refers to one of the two regions formed when a line divides a two-dimensional plane. It is essential for understanding linear equations and inequalities, as it represents the solutions to those equations or inequalities. In particular, when dealing with linear inequalities, the half-plane can indicate all the possible values that satisfy the inequality, extending infinitely in one direction.
Intersection: The intersection of two or more sets is the set containing all elements that are common to each of the sets. This concept is crucial for understanding relationships between different groups, helping visualize shared traits or properties through various methods.
Intersection of two sets: The intersection of two sets is a new set containing all the elements that are common to both original sets. The symbol for intersection is ∩.
Linear Inequality: A linear inequality is a mathematical expression that shows the relationship between two values where one value is not equal to the other, typically expressed in the form of $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$. These inequalities can represent ranges of values rather than single points, and they are used to model situations where constraints exist, making them essential in understanding how to evaluate and compare different scenarios.
Linear programming: Linear programming is a mathematical technique used to maximize or minimize a linear objective function, subject to a set of linear constraints. It is widely used in fields such as economics, business, engineering, and military applications.
Linear programming: Linear programming is a mathematical technique used to optimize a linear objective function, subject to a set of linear constraints, usually in the form of inequalities. This method helps in determining the best possible outcome, such as maximizing profit or minimizing costs, while adhering to given limitations like resources or budget. By visualizing constraints as linear inequalities in a graph, solutions can be found at the vertices of the feasible region created by these inequalities.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, typically representing the quantity to be maximized or minimized. This function plays a central role in determining the best outcome based on constraints and resources available. By evaluating different variable values, the objective function helps identify optimal solutions within feasible regions defined by linear inequalities.
Optimization problem: An optimization problem is a mathematical situation where the goal is to find the best solution from a set of feasible solutions, often defined by constraints and objectives. In the context of linear inequalities, these problems involve maximizing or minimizing a linear objective function subject to constraints represented as inequalities. The feasible region created by these inequalities helps in identifying the optimal solution.
Ordered Pair: An ordered pair is a pair of elements where the order of the elements matters, typically represented as (x, y) in a coordinate system. The first element, x, is known as the x-coordinate, and the second element, y, is known as the y-coordinate. This concept is crucial for representing points in a two-dimensional space, which directly connects to solving equations and inequalities involving two variables.
Quadrants: Quadrants are the four sections of a Cartesian coordinate system created by the intersection of the x-axis and y-axis. Each quadrant is designated by a number (I, II, III, IV) and contains specific combinations of positive and negative values for x and y coordinates. Understanding quadrants is essential for graphing linear equations, functions, and systems of inequalities since they determine where points lie in the coordinate plane.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.
Solid boundary line: A solid boundary line is a type of line used in the graph of a system of linear inequalities that indicates that the points on the line are included in the solution set. This line represents the equations of the inequalities and differentiates between solutions that satisfy the inequality and those that do not. It is crucial for understanding how to graphically represent and interpret the solutions to linear inequalities in two variables.
Solution set: A solution set is the collection of all values that satisfy a given equation or inequality. In mathematical contexts, this term is crucial as it helps identify all possible answers that make an equation true or meet the conditions of an inequality, creating a clearer understanding of relationships between variables.
System of linear inequalities: A system of linear inequalities is a set of two or more inequalities that involve the same variables, representing constraints on those variables. These inequalities define a region in the coordinate plane, known as the feasible region, where all solutions to the system can be found. The solutions to a system of linear inequalities can be represented graphically, and they often overlap, creating a multi-dimensional space where various conditions are satisfied simultaneously.
Y-intercept: The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.