Compound Inequalities

A compound inequality are when two or more inequalities joined together with either “and” or “or.”
 
In this lesson, we will look at how to find and graph the solution set for two compound inequalities.

 

Remember :

A compound inequality is when you have 2 or more inequalities and you want to find the solution set for the problem.

Lets look other ways a compound inequality may be written.

    Example 1.     $ \color{blue}{14 \leq 2n+18 \lt 20} $
    Example 2.     $ \color{#800080}{2a+3 \gt 7}$    $\color{#800080}{\cap}$    $\color{#800080} {2a+9 \geq 11} $
    Example 3.     $ \color{#008080}{12 \gt {^-3y}}$    $\color{#008080}{\cup}$    $\color{#008080}{ {^-2y} \gt {^-12}} $

 

Solutions:

Example 1.     $ \color{blue}{14 \leq 2n+18 \lt 20} $

This inequality can be broken down into 2 separate inequalities:
\[\begin{align*}
\color{blue}{14}&\color{blue}{\leq 2n+18} \lt 20 & 14 \leq \color{blue}{2n+18} &{\color{blue}{\lt20}}\\
14 &\leq 2n+18 & 2n+18 &\lt20
\end{align*}\]
The solution set will be the intersection of these two inequalities.

    $14 \leq 2n+18$    and    $ 2n+18 \lt 20 $
    $14 \color{blue}{-18} \leq 2n+18 \color{blue}{-18}$    and    $2n+18 \color{blue}{-18} \lt 20 \color{blue}{-18}$
    $-4 \leq 2n$    and    $ 2n \lt 2$
    $\color{blue}{-2 \leq n}$ or    and     $ \color{blue}{n \lt 1}$
       $\color{blue}{n \geq -2}$  
    n is greater than -2    and     n is less than 1

 
So the solution set can be written as $\color{blue}{-2 \leq n \lt 1}$
and can be graphed on a number line:
Graph of n is greater than -2 and less than or equal to 1
  
 
 
green-and 
 

Example 2.     $ \color{#800080}{2a+3 \gt 7}$    $\color{#800080}{\cap}$    $\color{#800080} {2a+9 \geq 11}$

The $\color{#800080}{\cap}$ represent the intersection of the two solution sets,
So we can replace the $\color{#800080}{\cap}$ with the word “and”.

      $2a+3 \gt 7$    and    $2a+9 \geq 11 $
      $2a+3 \color{#800080}{-3} \gt 7 \color{#800080}{-3}$    and    $2a+9 \color{#800080}{-9} \geq 11 \color{#800080}{-9}$
      $2a \gt 4$    and    $2a \geq 2$
       $\color{#800080}{a \gt 2}$    and     $ \color{#800080}{a \geq 1}$
       a is greater than 2    and     a is greater than or equal to 1

 
So the solution set to this problem is the overlap of the 2 individual graphs, $\color{#800080}{a \gt 2}$
and can be graphed on a number line:
graphing inequalities: a is greater than 2
a union is the same as or  
 
 
 

Example 3.     $ \color{#008080}{12 \gt {^-3y}}$    $\color{#008080}{\cup}$    $\color{#008080}{ {^-2y} \gt {^-12}} $

When a compound inequality has a $\color{#008080}{\cup}$ between the 2 inequalities the $\color{#008080}{\cup}$ represent the union of the two solution sets, so we can replace the $\color{#008080}{\cup}$ with the word “or”.

Remember when you multiply or divide an inequality by a negative number you flip the sign.

      $12 \gt -3y $    or    $-2y \gt -12 $
      $\frac {12}{\color{#008080}{-3}} \color{#008080}{\lt} \frac{-3y } {\color{#008080}{-3}} $    or    $\frac {-2y}{\color{#008080}{-2}} \color{#008080}{\lt} \frac{-12} {\color{#008080}{-3}} $
       $\color{#008080}{y \lt -4}$    or     $ \color{#008080}{y \lt 6}$
       y is less than -4    or     y is less than 6

 
So the solution set is the combination of the 2 individual graphs, $\color{#008080}{y \lt 6}$
and can be graphed on a number line:
The union is y is less than 6