Absolutely, I’d be happy to help you understand interval notation! Interval notation is a way to represent a set of numbers that fall within a certain range or interval. It is often used in mathematics to describe domains and ranges, as well as solutions to inequalities.
Basic Concepts:
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Open Intervals: An open interval does not include its endpoints. It's represented using parentheses.
- Example: (2, 5) means all numbers greater than 2 and less than 5, but not including 2 and 5.
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Closed Intervals: A closed interval includes its endpoints. It's represented using brackets.
- Example: [2, 5] means all numbers from 2 to 5, including 2 and 5.
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Half-Open (or Half-Closed) Intervals: These include one endpoint but not the other.
- Example: [2, 5) means all numbers from 2 to 5, including 2 but not including 5. (2, 5] does the opposite.)
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Negative and Positive Infinity: Infinity (∞) is always treated as an open interval because it cannot be reached. You use parentheses with infinity.
- Example: (−∞, 3) means all numbers less than 3 (3 not included).
- Example: (4, ∞) means all numbers greater than 4 (4 not included).
Examples:
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Example 1: \( x > 3 \): In interval notation, this is expressed as \( (3, \infty) \).
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Example 2: \( x \leq 7 \): In interval notation, this is expressed as \( (-\infty, 7] \).
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Example 3: \( 2 < x < 5 \): In interval notation, this is expressed as \( (2, 5) \).
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Example 4: \( x \geq 0 \) and \( x < 4 \): In interval notation, this is expressed as \( [0, 4) \).
Combining Intervals:
Sometimes, you may need to combine intervals if the solution does not form a single continuous range.
- Example 5: For \( x < -1 \) or \( x \geq 2 \), you would write this as \( (-\infty, -1) \cup [2, \infty) \).
Practice:
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Translate the following to interval notation:
- \( x \leq 3 \): \( (-\infty, 3] \)
- \( x > 1 \) and \( x < 4 \): \( (1, 4) \)
- \( x = 0 \): \( {0} \) (this is technically not interval notation, but you might write it as \( [0, 0] \) in some contexts)
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Write the following in standard notation:
- \( [2, 5) \): represents the range including 2 but not including 5.
- \( (-\infty, 0) \cup (0, \infty) \): represents all real numbers except 0.
Summary:
- Use parentheses
()for open intervals (not including endpoints). - Use brackets
[]for closed intervals (including endpoints). - Use negative/positive infinity with parentheses only.
Understanding and using interval notation takes practice, so try translating different inequalities into interval notation to strengthen your understanding. Good luck on your exam!