Compound Inequalities: Intersection and Union of Sets

Section 2.5: Compound Inequalities

Objective 1: Intersection of Two Sets

Compound inequalities are formed when two inequalities are joined by the words and or or. Understanding the intersection of sets is essential for solving compound inequalities involving 'and'.

• Compound Inequality: An inequality that combines two inequalities using 'and' or 'or'.

• Intersection: The intersection of two sets, A and B, is the set of all elements common to both sets. It is denoted as $A \cap B$.

• Notation: $A \cap B$

• Example: If $A = \{x \mid x \text{ is an even integer}\}$ and $D = \{4, 5, 6, 7\}$, then $A \cap D = \{4, 6\}$.

Objective 2: Solving Compound Inequalities with 'and'

When two inequalities are joined by 'and', the solution set is the intersection of the solution sets of each inequality. The values must satisfy both inequalities simultaneously.

• Solving Steps:

  1. Solve each inequality separately.

  2. Find the intersection of the solution sets.

  3. Express the solution in interval notation.

• Example 1: Solve $x + 1 \geq 7$ and $3x - 1 \geq 5$.

  • $x + 1 \geq 7 \Rightarrow x \geq 6$

  • $3x - 1 \geq 5 \Rightarrow 3x \geq 6 \Rightarrow x \geq 2$

  • Intersection: $x \geq 6$ (since $x$ must satisfy both, the more restrictive condition applies)

  • Interval notation: $[6, \infty)$

• Example 2: Solve $2 \leq 3x + 1 \leq 4$.

  • Break into two inequalities:

    • $2 \leq 3x + 1$

    • $3x + 1 \leq 4$

  • Solve $2 \leq 3x + 1 \Rightarrow 1 \leq 3x \Rightarrow \frac{1}{3} \leq x$

  • Solve $3x + 1 \leq 4 \Rightarrow 3x \leq 3 \Rightarrow x \leq 1$

  • Intersection: $\frac{1}{3} \leq x \leq 1$

  • Interval notation: $\left[\frac{1}{3}, 1\right]$

Objective 3: Union of Two Sets

Compound inequalities joined by 'or' require finding the union of the solution sets. The union includes all elements that belong to either set.

• Union: The union of two sets, A and B, is the set of all elements that belong to either set. It is denoted as $A \cup B$.

• Notation: $A \cup B$

• Example: If $C = \{2, 3, 4, 5\}$ and $D = \{4, 5, 6, 7\}$, then $C \cup D = \{2, 3, 4, 5, 6, 7\}$.

Set	Elements
C	2, 3, 4, 5
D	4, 5, 6, 7
C \cup D	2, 3, 4, 5, 6, 7

Objective 4: Solving Compound Inequalities with 'or'

For compound inequalities joined by 'or', the solution set is the union of the solution sets of each inequality. Values that satisfy either inequality are included.

• Solving Steps:

  1. Solve each inequality separately.

  2. Find the union of the solution sets.

  3. Express the solution in interval notation.

• Example: Solve $3x + 2 \leq 5$ or $7x > 29$.

  • $3x + 2 \leq 5 \Rightarrow 3x \leq 3 \Rightarrow x \leq 1$

  • $7x > 29 \Rightarrow x > \frac{29}{7}$

  • Union: $x \leq 1$ or $x > \frac{29}{7}$

  • Interval notation: $(-\infty, 1] \cup \left(\frac{29}{7}, \infty\right)$

Summary Table: Intersection vs. Union

Operation	Symbol	Description	Compound Inequality
Intersection	$\cap$	Elements common to both sets	Joined by 'and'
Union	$\cup$	Elements in either set	Joined by 'or'

Key Terms: Compound inequality, intersection, union, interval notation.

Applications: Compound inequalities are used to describe ranges of values in mathematics, science, and engineering, such as temperature ranges, acceptable measurements, or solution sets for equations.