Glossary

4.7 Graphing Systems of Linear Inequalities

3 min read•june 24, 2024

Linear inequalities are powerful tools for modeling real-world scenarios with constraints. They allow us to visualize and solve complex problems by graphing multiple inequalities on a . The solution set is where all shaded regions overlap.

Graphing systems of linear inequalities involves plotting each inequality separately and identifying the overlapping area. This method helps solve problems in fields like economics and engineering, where we need to find the best solution within given constraints.

Graphing Systems of Linear Inequalities

Solutions of linear inequality systems

Top images from around the web for Solutions of linear inequality systems

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Solutions of linear inequality systems

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

  • Graph Solutions to Systems of Linear Inequalities | Intermediate Algebra View original

    Is this image relevant?

1 of 3
  • Consist of two or more linear inequalities sharing the same variables
    • Each inequality represented by a line and on a coordinate plane
    • Solution set is the region where all shaded areas overlap
  • Identify solutions by finding points (x,y)(x, y) satisfying all inequalities simultaneously
    • These points lie in the region where all shaded areas intersect
  • Solution set can be bounded (closed) or unbounded (open)
    • Bounded: solution region is a closed polygon with finite area (triangle, square)
    • Unbounded: solution region extends infinitely in one or more directions (, quadrant)

Graphing of inequality systems

  • Graph each linear inequality separately on the same coordinate plane
    • Represent using for ≤\leq or ≥\geq and for [<](https://www.fiveableKeyTerm:<)[<](https://www.fiveableKeyTerm:<) or [>](https://www.fiveableKeyTerm:>)[>](https://www.fiveableKeyTerm:>)
    • Shade the region satisfying the inequality
      • If yy isolated, shade above line for >> or ≥\geq and below for << or ≤\leq
      • If xx isolated, shade right of line for >> or ≥\geq and left for << or ≤\leq
  • Solution set is the region where all shaded areas overlap
    • Identify points (x,y)(x, y) lying within this overlapping region
  • Label solution set using set-builder or
    • : {(x,y)∣x≥0,y≥0,y≤2x+1}\{(x, y) \mid x \geq 0, y \geq 0, y \leq 2x + 1\}
    • Interval notation: x≥0,y∈[0,2x+1]x \geq 0, y \in [0, 2x + 1]

Components of Linear Inequalities

  • form the basis of linear inequalities
    • Expressed in the form y=mx+by = mx + b, where mm represents the slope
    • Slope indicates the rate of change between variables
  • Intercepts are points where the line crosses the x or
    • x-: point where line intersects (x,0)(x, 0)
    • y-intercept: point where line intersects y-axis (0,y)(0, y)

Real-world applications of inequalities

  • Identify variables and constraints in the problem
    • Variables typically represent quantities (cost, revenue, production)
    • Constraints are limitations or requirements, expressed as inequalities (budget, demand, supply)
  • Formulate the based on given constraints
    • Each constraint translates to a linear inequality
  • Graph the system of linear inequalities to visualize solution set
    • Solution set represents satisfying all constraints
  • Interpret solution set in context of the problem
    • Identify points (x,y)(x, y) within feasible region optimizing desired outcome
    • Maximize profit or minimize cost while meeting production requirements (resource allocation, production planning)
  • Express solution using appropriate units and labels based on problem context
  • Optimization problems often involve finding the best solution within the feasible region

Key Terms to Review (28)

<: The less than symbol, <, is a mathematical operator that indicates a relationship where one value is smaller than another value. It is used in various contexts within algebra to represent inequalities, where the solution set includes all values that satisfy the inequality condition.
>: The greater than symbol (>) is a mathematical operator used to compare two values and indicate that one value is larger than the other. It is a fundamental concept in algebra that is applied in various contexts, including solving linear inequalities, compound inequalities, absolute value inequalities, graphing linear inequalities in two variables, graphing systems of linear inequalities, solving rational inequalities, and solving quadratic inequalities.
≤ (Less Than or Equal To): The symbol '≤' represents the mathematical relationship of 'less than or equal to'. It is used to compare two values and indicate that one value is less than or equal to the other value. This key term is essential in understanding and working with various mathematical concepts, including integers, linear inequalities, compound inequalities, absolute value inequalities, linear inequalities in two variables, systems of linear inequalities, rational inequalities, and quadratic inequalities.
Boundary Line: A boundary line is a conceptual dividing line that separates regions or areas based on certain criteria. In the context of graphing linear inequalities and systems of linear inequalities, the boundary line represents the line that separates the solutions that satisfy the inequality from those that do not.
Bounded Solution: A bounded solution refers to a solution set that is confined within a specific region or range, as opposed to an unbounded solution that can extend infinitely. In the context of graphing systems of linear inequalities, a bounded solution represents the feasible region where all the constraints are satisfied simultaneously.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional graphical representation used to locate and visualize points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a point called the origin, forming a grid-like structure that allows for the precise mapping of coordinates.
Dashed Line: A dashed line is a type of line that is composed of a series of short line segments separated by gaps, rather than a continuous unbroken line. This visual representation is used in various mathematical and graphical contexts to convey specific meanings or properties.
Feasible Region: The feasible region is the set of all possible solutions that satisfy a system of linear inequalities in two variables. It represents the area on a coordinate plane where all the constraints or inequalities are met simultaneously.
Graphical Method: The graphical method is a visual approach to solving mathematical problems, where the solution is obtained by interpreting and analyzing the graphical representation of the problem. This method is particularly useful in the context of solving absolute value inequalities, graphing systems of linear inequalities, and finding composite and inverse functions.
Greater Than or Equal To (≥): The symbol ≥ is a mathematical operator that represents the relationship where one value is greater than or equal to another value. It is used to compare quantities and express inequalities, indicating that the left-hand side is either greater than or equal to the right-hand side.
Half-plane: A half-plane is a region of the coordinate plane that is divided by a line. It represents all the points on one side of the line, including the line itself. This concept is particularly important in the context of graphing linear inequalities in two variables and systems of linear inequalities.
Infinite Solutions: Infinite solutions refers to a situation where a linear equation or a system of linear equations has an unlimited number of solutions. This concept is particularly relevant in the context of solving linear equations, solving applications with systems of equations, and graphing systems of linear inequalities.
Intercept: The intercept of a function or a line is the point where the function or line intersects the y-axis. It represents the value of the function or the coordinate of the line when the independent variable (typically x) is zero.
Intersection: The intersection of two or more sets refers to the elements that are common to all of those sets. It represents the overlap or shared points between the sets.
Interval Notation: Interval notation is a way to represent a range of numbers or values using a specific set of symbols and conventions. It is commonly used to describe the solutions or solutions sets of various types of inequalities, as well as to graph and visualize these solutions on a number line.
Linear Equations: A linear equation is a mathematical equation in which the variables are raised to the first power and the equation can be represented as a straight line on a graph. These equations are fundamental in solving systems of equations and graphing systems of linear inequalities.
No Solution: The term 'no solution' refers to a situation in which an equation, system of equations, or system of linear inequalities does not have a valid solution that satisfies all the given constraints. This means that there are no values for the variables that can make the equation or system of equations/inequalities true.
Optimization: Optimization is the process of finding the best or most favorable solution to a problem, given certain constraints or objectives. It involves selecting the optimal values for decision variables to achieve the desired outcome, whether that is maximizing a benefit or minimizing a cost.
Origin: The origin is a fundamental concept in mathematics, particularly in the context of coordinate systems and graphing. It represents the fixed point of reference from which all other points are measured and located on a graph or coordinate plane.
Set-Builder Notation: Set-builder notation is a way to define a set by specifying the properties or characteristics that its elements must satisfy. It provides a concise and precise way to represent sets using mathematical symbols and logical statements.
Shaded Region: The shaded region refers to the area on a graph that represents the solution set for a linear inequality or a system of linear inequalities. It is a visual representation of the values that satisfy the given inequality or set of inequalities.
Solid Line: A solid line is a continuous, unbroken line used in graphing and visual representations to denote specific properties or relationships. In the context of linear inequalities and systems of linear inequalities, the solid line is a key element in accurately depicting the solution set.
System of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously. These systems are used to represent and solve real-world problems involving multiple constraints or conditions that need to be met at the same time.
Test Point: A test point is a specific location or value within a mathematical function or inequality where the function or inequality is evaluated to determine its behavior, such as the sign of the expression or whether it satisfies the given conditions.
Unbounded Solution: An unbounded solution in the context of graphing systems of linear inequalities refers to a solution set that is not confined to a specific region or bounded area on the coordinate plane. It indicates that the solution set extends indefinitely in one or more directions, without any finite boundaries.
Union: In mathematics, a union refers to the combination of two or more sets that includes all the elements from each set without duplication. This concept is crucial when dealing with inequalities, as it helps identify the total range of solutions that satisfy at least one of the conditions, allowing for a comprehensive understanding of overlapping and distinct solution sets.
X-axis: The x-axis is the horizontal line in a coordinate plane that represents the independent variable or the values along the horizontal dimension. It is the primary reference line for measuring the positions of points along the horizontal direction.
Y-axis: The y-axis is the vertical line on a coordinate plane that represents the vertical or up-and-down dimension. It is used to measure and plot the position of points along the vertical axis of a graph.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide