system of inequalities

(noun)

A set of inequalities with multiple variables, often solved with a particular specification of the values of all variables that simultaneously satisfies all of the inequalities.

Related Terms

  • mutually exclusive
  • subset

Examples of system of inequalities in the following topics:

  • Solving Systems of Linear Inequalities

    • A system of inequalities is a set of inequalities with multiple variables, often solved with a particular specification of of the values of all variables that simultaneously satisfies all of the inequalities.
    • A system of inequalities can be solved graphically and non-graphically.
    • Often the easiest way to solve a system of linear inequalities is by graphing.
    • If all of the inequalities of a system fail to overlap over the same area, then there is no solution to that system.
    • There is no area which is shaded by all three inequalities, so the system of inequalities has no solution.
  • Nonlinear Systems of Inequalities

    • Systems of nonlinear inequalities can be solved by graphing boundary lines.
    • A system of inequalities consists of two or more inequalities, which are statements that one quantity is greater than or less than another.
    • This area is the solution to the system.
    • The limits of each inequality intersect at $(-1, 1)$ and $(2, 4)$.
    • Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area.
  • Graphs of Linear Inequalities

    • The simplest inequality to graph is a single inequality in two variables, usually of the form: $y\leq mx+b$, where the inequality can be of any type, less than, less than or equal to, greater than, greater than or equal to, or not equal to.
    • To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.
    • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
    • This also means that if there are inequalities that don't overlap, then there is no solution to the system.
    • The brown overlapped shaded area is the final solution to the system of linear inequalities because it is comprised of all possible solutions to $y<-\frac{1}{2}x+1$ (the dotted red line and red area below the line) and $y\geq x-2$ (the solid green line and the green area above the line).  
  • Equations and Inequalities

    • In a set of simultaneous equations, or system of equations, multiple equations are given with multiple unknowns.
    • A solution to the system is an assignment of values to all the unknowns so that all of the equations are true.
    • An inequality is a relation that holds between two values when they are different.
    • These relations are known as strict inequalities.
    • In contrast to strict inequalities, there are two types of inequality relations that are not strict:
  • Linear Inequalities

    • This just means that you need to find the values of the variable that make the inequality true.  
    • There is only one rule that is different: When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.
    • Step 1, combine like terms on each side of the inequality symbol:
    • Step 2, since there is a variable on both sides of the inequality, choose to move the $-4x$, to combine the variables on the left hand side of the inequality.
    • Notice the open circle means that the value of $4$ in not a solution to the inequality since $4>4$ is a false statement.  
  • Compound Inequalities

    • Another type of inequality is the compound inequality, which can also be solved to find the possible values for a variable.
    • One type of inequality is the compound inequality.
    • A compound inequality is of the form:
    • Subtract 6 from all three parts of the inequality:
    • Solve a compound inequality by balancing all three components of the inequality
  • Rules for Solving Inequalities

    • Operations can be conducted on inequalities and used to solve inequalities for all possible values of a variable.
    • Any value $c$ may be added to or subtracted from both sides of an inequality:
    • Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality.
    • Solving an inequality gives all of the possible values that the variable can take to make the inequality true.
    • Recognize how operations on an inequality affect the sense of the inequality
  • Solving Problems with Inequalities

    • These types of relationships are not relationships of equality but, rather, relationships of inequality.
    • Speculate on the number of solutions of a linear inequality.
    • If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged.
    • If both sides of an inequality are multiplied or divided by the same positive number, the sense of the inequality remains unchanged.
    • Miah was asked to find the values of x that make this inequality true: 2x + 1 ≤ 7.
  • Graphing Inequalities

    • The solutions to inequalities can be graphed by drawing a boundary line and shading half of the plane.
    • We now wish to determine the location of the solutions to linear inequalities in two variables.
    • Linear inequalities in two variables are inequalities of the forms:
    • The method of graphing linear inequalities in two variables is as follows:
    • Graph an inequality by shading the correct section of the plane
  • Introduction to Inequalities

    • In mathematics, inequalities are used to compare the relative size of values.
    • A description of different types of inequalities follows.
    • In the two types of strict inequalities, $a$ is not equal to $b$.
    • In contrast to strict inequalities, there are two types of inequality relations that are not strict:
    • To see why, consider the left side of the inequality.
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.