Glossary

2.6 Solve Compound Inequalities

3 min read•june 24, 2024

Compound inequalities combine multiple conditions using "" "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 , 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 (≤)
  • The is the collection of all values that satisfy an inequality or

Compound inequalities with "and"

  • Represent the 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 (bounded by parentheses for exclusive endpoints and brackets for inclusive endpoints) or graphically on a
  • Example: 2<x≤62 < x \leq 6 and 4≤x<84 \leq x < 8
    • Solve 2<x≤62 < x \leq 6 and 4≤x<84 \leq x < 8 independently
    • Identify the common region satisfying both inequalities: 4≤x≤64 \leq x \leq 6
    • Solution set in interval notation: [4,6][4, 6]

Compound inequalities with "or"

  • Represent the 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<−1x < -1 or x>3x > 3
    • Solve x<−1x < -1 and x>3x > 3 independently
    • Combine the solution sets: x<−1x < -1 or x>3x > 3
    • Solution set in interval notation: (−∞,−1)∪(3,∞)(-\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: pp (product weight) and ww (packaging weight)
    • Constraints: p+w≤2p + w \leq 2 and p>1.5p > 1.5
    • Compound inequality: p>1.5p > 1.5 and p+w≤2p + w \leq 2
    • Solve the compound inequality:
      1. p>1.5p > 1.5
      2. w≤2−pw \leq 2 - p, substituting the lower bound of pp: w≤2−1.5=0.5w \leq 2 - 1.5 = 0.5
    • Interpretation: The packaging weight must be less than or equal to 0.5 pounds

Key Terms to Review (21)

<: The less than symbol, <, is a mathematical operator that indicates a relationship where one value is smaller than another value. It is used in various contexts within algebra to represent inequalities, where the solution set includes all values that satisfy the inequality condition.
>: The greater than symbol (>) is a mathematical operator used to compare two values and indicate that one value is larger than the other. It is a fundamental concept in algebra that is applied in various contexts, including solving linear inequalities, compound inequalities, absolute value inequalities, graphing linear inequalities in two variables, graphing systems of linear inequalities, solving rational inequalities, and solving quadratic inequalities.
∪: The union symbol, ∪, represents the operation of combining two or more sets into a single set that contains all the unique elements from the original sets. It is a fundamental concept in set theory and is particularly relevant in the context of solving compound inequalities.
≤ (Less Than or Equal To): The symbol '≤' represents the mathematical relationship of 'less than or equal to'. It is used to compare two values and indicate that one value is less than or equal to the other value. This key term is essential in understanding and working with various mathematical concepts, including integers, linear inequalities, compound inequalities, absolute value inequalities, linear inequalities in two variables, systems of linear inequalities, rational inequalities, and quadratic inequalities.
Algebraic Method: The algebraic method refers to the use of algebraic principles, equations, and techniques to solve problems and find solutions. It involves the application of mathematical operations and logical reasoning to manipulate variables and expressions in order to arrive at a desired result.
And: The term 'and' is a logical conjunction used to connect two or more related statements, ideas, or conditions. In the context of solving compound inequalities, 'and' is used to represent the intersection of two or more inequalities, where the solution must satisfy all the given conditions simultaneously.
Bounded Interval: A bounded interval is a set of real numbers that has both a lower and an upper bound, meaning it is limited on both ends. This concept is particularly important in the context of solving compound inequalities, where the goal is to find the range of values that satisfy multiple constraints.
Compound Inequality: A compound inequality is a statement that involves two or more simple inequalities combined using the logical connectives 'and' or 'or'. It represents a range of values that satisfy all the individual inequalities within the compound statement.
Critical Points: Critical points refer to the specific values of a function where the derivative of the function is equal to zero or undefined. These points are crucial in analyzing the behavior and characteristics of a function, such as local maxima, local minima, and points of inflection.
Graphing Method: The graphing method is a fundamental technique used to visually represent and analyze mathematical relationships, such as the solutions to equations, inequalities, and systems of equations. This method involves plotting points on a coordinate plane to depict the behavior and characteristics of these mathematical expressions.
Greater Than or Equal To (≥): The symbol ≥ is a mathematical operator that represents the relationship where one value is greater than or equal to another value. It is used to compare quantities and express inequalities, indicating that the left-hand side is either greater than or equal to the right-hand side.
Inequality Symbols: Inequality symbols are mathematical notations used to represent relationships between two quantities or expressions where one is greater than, less than, or not equal to the other. These symbols are essential in solving linear and compound inequalities.
Intersection: The intersection of two or more sets refers to the elements that are common to all of those sets. It represents the overlap or shared points between the sets.
Intersection (∩): The intersection, denoted by the symbol ∩, is a set operation that describes the common elements between two or more sets. It represents the set of all elements that belong to both (or all) of the given sets simultaneously.
Interval Notation: Interval notation is a way to represent a range of numbers or values using a specific set of symbols and conventions. It is commonly used to describe the solutions or solutions sets of various types of inequalities, as well as to graph and visualize these solutions on a number line.
Number Line: The number line is a visual representation of the set of real numbers, extending infinitely in both the positive and negative directions. It serves as a fundamental tool in understanding and working with various mathematical concepts, including integers, linear inequalities, compound inequalities, rational inequalities, and quadratic inequalities.
Or: The logical operation 'or' is a fundamental concept in mathematics and computer science, representing a condition where at least one of the given statements or conditions must be true for the overall statement to be true.
Set-Builder Notation: Set-builder notation is a way to define a set by specifying the properties or characteristics that its elements must satisfy. It provides a concise and precise way to represent sets using mathematical symbols and logical statements.
Solution Set: The solution set is the set of all values of the variable(s) that satisfy an equation, inequality, or system of equations or inequalities. It represents the collection of all possible solutions to a given mathematical problem.
Unbounded Interval: An unbounded interval is a set of real numbers that extends infinitely in one or both directions on the number line, without a defined endpoint. It is a type of interval that is not limited by a specific upper or lower bound.
Union: In mathematics, a union refers to the combination of two or more sets that includes all the elements from each set without duplication. This concept is crucial when dealing with inequalities, as it helps identify the total range of solutions that satisfy at least one of the conditions, allowing for a comprehensive understanding of overlapping and distinct solution sets.
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