Compound Inequalities (Sets) Notes

Overview of Compound Inequalities

• Teacher Name: McGooden

• Notes Number: #4

• Subject: Compound Inequalities (Sets)

Definition

• Compound Inequality: A compound inequality is formed when two individual inequalities are joined together using a logical connective word, specifically and or or.

Key Concepts and Notations

• Conjunction ("And" Inequalities):

  • Expresses a set of numbers that satisfy both inequality conditions simultaneously.

  • Graphically represented as an overlap or bounded segment between two boundary values.

  • Expressed algebraically in bounded form: $-2 < x \le 6$.

• Disjunction ("Or" Inequalities):

  • Expresses a set of numbers that satisfy at least one of the inequality conditions.

  • Graphically represented as two separate rays pointing in opposite directions away from boundary values.

  • Expressed in set notation using the union symbol $\cup$.

• Boundary Point Rules:

  • Open Circle (Exclusive Boundary): Used for strict inequalities ($>$ or $<$). Corresponds to parentheses $($ or $)$ in interval notation.

  • Closed / Solid Circle (Inclusive Boundary): Used for inclusive inequalities ($\ge$ or $\le$). Corresponds to square brackets $[$ or $]$ in interval notation.

• Translating Key Phrasings into Inequality Symbols:

  • "Greater than": $>$

  • "Less than": $<$

  • "Exceeds": $>$

  • "At most": $\le$

  • "At least": $\ge$

  • "Maximum of": $\le$

Worked Examples

Example 1: Conjunction ("And") Inequality

• Problem: Graph a number greater than $-2$ and at most $6$.

• Translation of Components:

  • "Greater than $-2$": $x > -2$

  • "At most $6$": $x \le 6$ (includes equality)

• Combined Compound Inequality:

  • $-2 < x \le 6$

• Interval Notation:

  • $(-2, 6]$

• Graph Description:

  • Open circle located at $-2$

  • Closed (filled) circle located at $6$

  • Continuous line segment connecting $-2$ and $6$

Example 2: Disjunction ("Or") Inequality

• Problem: Graph a number less than $0$ or at least $5$.

• Translation of Components:

  • "Less than $0$": $x < 0$

  • "At least $5$": $x \ge 5$

• Combined Compound Inequality:

  • $x < 0 \quad \text{or} \quad x \ge 5$

• Interval Notation:

  • $(-\infty, 0) \cup [5, \infty)$

• Graph Description:

  • Open circle located at $0$ with an arrow pointing left toward $-\infty$

  • Closed (filled) circle located at $5$ with an arrow pointing right toward $\infty$

Example 3: Disjunction ("Or") Inequality with Exceeds and At Most

• Problem: Graph a number that exceeds $5$ or is at most $-1$.

• Translation of Components:

  • "Exceeds $5$": $x > 5$

  • "At most $-1$": $x \le -1$

• Combined Compound Inequality:

  • $x \le -1 \quad \text{or} \quad x > 5$

• Interval Notation:

  • $(-\infty, -1] \cup (5, \infty)$

• Graph Description:

  • Closed (filled) circle located at $-1$ with an arrow pointing left toward $-\infty$

  • Open circle located at $5$ with an arrow pointing right toward $\infty$

Example 4: Conjunction ("And") Inequality with At Least and Maximum

• Problem: Graph a number that is at least $2$ and is a maximum of $10$.

• Translation of Components:

  • "At least $2$": $x \ge 2$

  • "Maximum of $10$": $x \le 10$

• Combined Compound Inequality:

  • $2 \le x \le 10$

• Interval Notation:

  • $[2, 10]$

• Graph Description:

  • Closed (filled) circle located at $2$

  • Closed (filled) circle located at $10$

  • Continuous line segment connecting $2$ and $10$

Example 5: Multi-Step Compound Inequality with Variable Solving

• Problem: Solve the inequality and graph:

  • $3T + 2 < -7 \quad \text{or} \quad -4T + 5 \le 1$

• Step-by-Step Solution for Left Inequality:

  • Given: $3T + 2 < -7$

  • Subtract $2$ from both sides:

  • $3T < -7 - 2$

  • $3T < -9$

  • Divide both sides by $3$:

  • $\frac{3T}{3} < \frac{-9}{3}$

  • $T < -3$

• Step-by-Step Solution for Right Inequality:

  • Given: $-4T + 5 \le 1$

  • Subtract $5$ from both sides:

  • $-4T \le 1 - 5$

  • $-4T \le -4$

  • Divide both sides by $-4$ (dividing by a negative number reverses the inequality sign from $\le$ to $\ge$):

  • $\frac{-4T}{-4} \ge \frac{-4}{-4}$

  • $T \ge 1$

• Combined Algebraic Solution:

  • $T < -3 \quad \text{or} \quad T \ge 1$

• Interval Notation:

  • $(-\infty, -3) \cup [1, \infty)$

• Graph Description:

  • Open circle located at $-3$ with an arrow pointing left toward $-\infty$

  • Closed (filled) circle located at $1$ with an arrow pointing right toward $\infty$