4.5       Compound Inequalities
Homework: EOO 1-61, 75

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DSPM 0800 Class Notes

A Compound Inequality is two or more inequalities joined by the word “and” or by the word “or”.

Joining Word	Symbol	Sets	Inequalities	Solution Set
AND		INTERSECTION	CONJUNCTION	Only elements that the sets have in common
OR		UNION	DISJUNCTION	All elements that are in either set

SETS:
Example: An intersection is only the elements that the sets have in common.

                       

Example: A union is all elements that are in either set.

                       

Example: The empty set is the set having NO elements.

                       


COMPOUND INEQUALITIES:
Example: The word “and” means intersection, which is only the elements that the graphs have in common.

                       
                        Graph of each inequality                              

                                       

                Graph in RED shows intersection

           

Example: The word “or” means union, which is all elements that are in either graph.

                       

  Graph of each inequality                              
Graph in RED shows union                   


Example: Solve each inequality, then find the union.

           
                                                                        Graph of each inequality                                                                                 

Graph in RED shows union


We see by the example above, that sometimes the graph of a union matches the graph showing each of the inequalities.

Example:  A three-part inequality is an abbreviated form of writing an intersection with the word “and”. A three-part inequality clearly shows the boundaries of x.

                       
Notice the function notation tells us to replace f(x) with 3x – 1.

                       

The solution set is between 1 on the left and 3 on the right as shown in graph below.    

            Graph in BLUE shows the interval of numbers in the solution set

 


Example: Use the accompanying graph of  y = 4 – x  to solve  4 – x  <  -2  or  4 – x  > 7.

Draw a horizontal line through –2 and a horizontal line through 7.

Where does the red line intersect the horizontal line y = -2?   At x = 6.
When are the y-values on the red line less than –2?    When x > 6.

 

Where does the red line intersect the horizontal line y = 7?   At x = -3.
When are the y-values on the red line greater than 7?   When x < -3.

                       

 

The union is the same as the graphs of each of the inequalities since the graphs are disjoint.