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Flashcards
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Visual Learning
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Study Notes
Interval Notation
- Interval notation uses parentheses and brackets to represent a set of real numbers
- Parentheses indicate an endpoint is not included
- Brackets indicate an endpoint is included
- Infinity symbols are always paired with parentheses
Types of Intervals
- Open interval: (a, b) represents all values between a and b, but not including a or b. {x | a < x < b}
- Closed interval: [a, b] represents all values between a and b, including a and b. {x | a ≤ x ≤ b}
- Half-open (or half-closed) interval: [a, b) represents all values between a and b, including a but not b. {x | a ≤ x < b} or (a, b] represents all values between a and b, including b but not a. {x | a < x ≤ b}
- Disjoint interval: (-∞, a) ∪ (b, ∞) represents all values less than a or greater than b. {x | x < a or x > b}
- Infinite interval: (-∞, a) represents all real numbers less than a, (-∞, a] represents all real numbers less than or equal to a, (a, ∞) represents all real numbers greater than a, [a, ∞) represents all real numbers greater than or equal to a;
- All real numbers: (-∞, ∞) represents the set of all real numbers. {x| x is a real number}
Solving Linear Inequalities
- Distributive Property: Use the distributive property to clear parentheses.
- Clear fractions/decimals: Multiply each side of the inequality by a common denominator to eliminate fractions.
- Combine like terms: Combine all the like terms on each side of the equation.
- Isolate variable terms: Use the addition property of inequality to get all variable terms on one side.
- Isolate the variable: Use the multiplication property of inequality to solve for the variable.
- Reverse inequality: When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
Solving Compound Inequalities
- "And" Compound Inequalities: Solve each inequality separately, then find the intersection of the solution sets to determine the values that satisfy both inequalities.
- "Or" Compound Inequalities: Solve each inequality separately, then find the union of the solution sets to determine the values that satisfy at least one of the inequalities.
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Description
This quiz covers the concepts of interval notation, including the different types of intervals such as open, closed, half-open, disjoint, and infinite intervals. Learn how to properly represent sets of real numbers using parentheses and brackets. Test your understanding of this fundamental topic in mathematics.
