compound inequality

Examples of compound inequality in the following topics:

• Compound Inequalities

  • A compound inequality involves three expressions, not two, but can also be solved to find the possible values for a variable.
  • The compound inequality $a < x < b$ indicates "betweenness"—the number $x$ is between the numbers $a$ and $b$.
  • For a visualization of this inequality, refer to the number line below.
  • Finally, it is customary (though not necessary) to write the inequality so that the inequality arrows point to the left (i.e., so that the numbers proceed from smallest to largest):
  • Solve a compound inequality by balancing all three components of the inequality

• Nonlinear Systems of Inequalities

  • A system of inequalities consists of two or more inequalities, which are statements that one quantity is greater than or less than another.
  • A nonlinear inequality is an inequality that involves a nonlinear expression—a polynomial function of degree 2 or higher.
  • The most common way of solving one inequality with two variables $x$ and $y$ is to shade the region on a graph where the inequality is satisfied.
  • Every inequality has a boundary line, which is the equation produced by changing the inequality relation to an equals sign.
  • If we have two inequalities, therefore, we shade in the overlap region, where both inequalities are simultaneously satisfied.

• Linear Inequalities

  • A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
  • For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable.
  • A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign.  
  • Solving the inequality is the same as solving an equation.  
  • Step 1, combine like terms on each side of the inequality symbol:

• Graphs of Linear Inequalities

  • Graphing linear inequalities involves graphing the original line, and then shading in the area connected to the inequality.
  • Graphing an inequality is easy.
  • First, graph the inequality as if it were an equation.
  • Now if there is more than one inequality, start off by graphing them one at a time, just as was done with a single inequality.
  • To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.

• Solving Problems with Inequalities

  • Speculate on the number of solutions of a linear inequality.
  • (Hint: Consider the inequalities x < x−6 and x ≥ 9. )
  • 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.
  • If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality.

• 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.
  • Again, draw all the inequalities and shade in the area that each inequality covers.
  • If there is no intersection, then the two inequalities are either mutually exclusive, or one of the inequalities is a subset of the other.
  • Three inequalities are graphed.
  • There is no area which is shaded by all three inequalities, so the system of inequalities has no solution.

• Graphing Inequalities

  • First, consider the inequality as an equation (i.e., replace the inequality sign with an equals sign) and graph that equation.
  • If the inequality is $<$ or $>$, draw the boundary line dotted.
  • To do so, consider the inequality as an equation:
  • Let's substitute $(0, 0)$ into the original inequality:
  • Graph showing all possible solutions of the given inequality.

• Rules for Solving Inequalities

  • Arithmetic operations can be 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 that includes a variable gives all of the possible values that the variable can take that make the inequality true.
  • Recognize how operations on an inequality affect the sense of the inequality

• Introduction to Inequalities

  • In mathematics, inequalities are used to compare the relative size of values.
  • A description of different types of inequalities follows.
  • The strict inequality symbols are $<$ and $>$.
  • In contrast to strict inequalities, there are two types of inequality relations that are not strict:
  • Note that an open circle is used if the inequality is strict (i.e., for inequalities using $>$ or $<$), and a filled circle is used if the inequality is not strict (i.e., for inequalities using $\geq$ or $\leq$).

• Polynomial Inequalities

  • Polynomials can be expressed as inequalities, the solutions for which can be determined from the polynomial's zeros.
  • Like any other function, a polynomial may be written as an inequality, giving a large range of solutions.
  • The best way to solve a polynomial inequality is to find its zeros.
  • This knowledge can then be used to determine the solutions of the inequality.  
  • Solve for the zeros of a polynomial inequality to find its solution