2.6 Solve Compound Inequalities

Compound inequalities combine multiple conditions using "and" or "or" to create more complex mathematical statements. They're essential for describing ranges and relationships between quantities in various real-world scenarios.

Mastering compound inequalities involves understanding inequality symbols, solution sets, and how to solve and interpret them. This skill is crucial for modeling and solving problems in fields like economics, engineering, and data analysis.

Compound Inequalities

Inequality Symbols and Solution Sets

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• Inequality symbols represent relationships between quantities:
  • Less than (<)
  • Greater than (>)
  • Less than or equal to (≤)
  • Greater than or equal to (≥)
• The solution set is the collection of all values that satisfy an inequality or compound inequality

Compound inequalities with "and"

• Represent the intersection of two or more inequalities where all conditions must be satisfied simultaneously
• Solve each inequality separately then identify the overlapping region that satisfies all inequalities
• Express the solution set using interval notation (bounded by parentheses for exclusive endpoints and brackets for inclusive endpoints) or graphically on a number line
• Example: $2 < x \leq 6$ and $4 \leq x < 8$
  • Solve $2 < x \leq 6$ and $4 \leq x < 8$ independently
  • Identify the common region satisfying both inequalities: $4 \leq x \leq 6$
  • Solution set in interval notation: $[4, 6]$

Compound inequalities with "or"

• Represent the union of two or more inequalities where at least one condition must be satisfied
• Solve each inequality separately then combine the solution sets of all inequalities
• Express the solution set using interval notation (use $\cup$ symbol for union) or graphically on a number line
• Example: $x < -1$ or $x > 3$
  • Solve $x < -1$ and $x > 3$ independently
  • Combine the solution sets: $x < -1$ or $x > 3$
  • Solution set in interval notation: $(-\infty, -1) \cup (3, \infty)$

Real-world applications of inequalities

• Identify the relevant variables and constraints described in the problem statement
• Translate the verbal descriptions into mathematical inequalities using "and" when all conditions must be met simultaneously or "or" when at least one condition must be satisfied
• Solve the resulting compound inequality to determine the feasible solution set
• Interpret the mathematical solution in the context of the original real-world problem
• Example: A manufacturer requires the total weight of a product and its packaging to be at most 2 pounds. The product itself must weigh more than 1.5 pounds. Find the allowable weight range for the packaging.
  • Variables: $p$ (product weight) and $w$ (packaging weight)
  • Constraints: $p + w \leq 2$ and $p > 1.5$
  • Compound inequality: $p > 1.5$ and $p + w \leq 2$
  • Solve the compound inequality:
    1. $p > 1.5$
    2. $w \leq 2 - p$, substituting the lower bound of $p$: $w \leq 2 - 1.5 = 0.5$
  • Interpretation: The packaging weight must be less than or equal to 0.5 pounds