Compound Inequalities: Intersection and Union of Sets

Chapter 2: Equations, Inequalities, and Problem Solving

Section 2.5: Compound Inequalities

This section explores compound inequalities, focusing on the intersection and union of sets as they relate to solving inequalities. Compound inequalities are essential in intermediate algebra for understanding how multiple conditions can be satisfied simultaneously or alternatively.

Objective 1: Finding the Intersection of Two Sets

The intersection of two sets is a fundamental concept in set theory and algebra. It is used to determine the solution set for compound inequalities joined by "and."

• Intersection of Sets: The intersection of sets A and B, denoted as A ∩ B, is the set of all elements common to both sets.

• Compound Inequality with "and": The solution set is the intersection of the solution sets of the individual inequalities.

• Notation: $A \cap B$

• Example: If A = {2, 4, 6} and B = {4, 6, 8}, then the intersection is {4, 6}.

Objective 2: Solving Compound Inequalities Containing “and”

Compound inequalities joined by "and" require that both conditions are satisfied. The solution is the overlap of the individual solution sets.

• Step 1: Solve each inequality separately.

• Step 2: Find the intersection of the solution sets.

• Step 3: Graph the solution, showing only the region where both inequalities are true.

• Important: When dividing by a negative number, reverse the direction of the inequality sign.

• Example: Solve $x + 4 > 0$ and $4x > 0$.

  • $x + 4 > 0 \Rightarrow x > -4$

  • $4x > 0 \Rightarrow x > 0$

  • Intersection: $x > 0$

Objective 3: Finding the Union of Two Sets

The union of two sets is used when solving compound inequalities joined by "or." The solution set includes all elements that satisfy at least one of the inequalities.

• Union of Sets: The union of sets A and B, denoted as A ∪ B, is the set of all elements that belong to either set.

• Compound Inequality with "or": The solution set is the union of the solution sets of the individual inequalities.

• Notation: $A \cup B$

• Example: If A = {2, 4, 6} and B = {3, 5, 8}, then the union is {2, 3, 4, 5, 6, 8}.

Objective 4: Solving Compound Inequalities Containing “or”

Compound inequalities joined by "or" require that at least one condition is satisfied. The solution is the union of the individual solution sets.

• Step 1: Solve each inequality separately.

• Step 2: Find the union of the solution sets.

• Step 3: Graph the solution, showing all regions where either inequality is true.

• Example: Solve $x < -2$ or $x > 3$.

  • Solution: $x < -2$ or $x > 3$

  • Union: All values less than -2 and all values greater than 3

Summary Table: Intersection vs. Union of Sets

Operation	Symbol	Definition	Example
Intersection	∩ ($\cap$)	Elements common to both sets	{4, 6} from A = {2, 4, 6}, B = {4, 6, 8}
Union	∪ ($\cup$)	Elements in either set	{2, 3, 4, 5, 6, 8} from A = {2, 4, 6}, B = {3, 5, 8}

Additional info: Compound inequalities are a key topic in intermediate algebra, providing foundational skills for solving more complex equations and understanding set relationships.