system of inequalities

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.