Write The Inequality In Interval Notation

Writing Inequalities in Interval Notation: A Comprehensive Guide

Inequalities are mathematical statements comparing two expressions, indicating that one is greater than, less than, greater than or equal to, or less than or equal to the other. Representing these inequalities in interval notation provides a concise and standardized way to express the solution set, which is crucial in various mathematical applications, especially in calculus and analysis. This guide will comprehensively cover writing inequalities in interval notation, exploring various types of inequalities and providing step-by-step examples to solidify your understanding.

Understanding Inequalities and Their Symbols

Before diving into interval notation, let's refresh our understanding of inequality symbols:

• >: Greater than
• <: Less than
• ≥: Greater than or equal to
• ≤: Less than or equal to
• ≠: Not equal to

These symbols form the basis of inequalities, which define a range of values rather than a single value like an equation. For example, x > 5 indicates that x can be any value greater than 5, while x ≤ -2 means x can be -2 or any value less than -2.

Introduction to Interval Notation

Interval notation offers a succinct method for representing solution sets of inequalities. It utilizes parentheses () and brackets [] to denote whether the endpoints are included or excluded from the interval.

• Parentheses (): Indicate that the endpoint is excluded from the interval. This is used for strict inequalities (> and <).

• Brackets []: Indicate that the endpoint is included in the interval. This is used for inequalities including equality (≥ and ≤).

• Infinity (∞) and Negative Infinity (-∞): These symbols represent unbounded intervals and are always accompanied by parentheses since infinity is not a number.

Different Types of Intervals and Their Notation

Let's explore the different types of intervals and their corresponding notation:

1. Bounded Intervals

These intervals have both a lower and upper bound.

• Open Interval (a, b): Represents all values between 'a' and 'b', excluding 'a' and 'b'. This corresponds to the inequality a < x < b. Example: (2, 5) represents all numbers between 2 and 5, but not 2 or 5 themselves.

• Closed Interval [a, b]: Represents all values between 'a' and 'b', including 'a' and 'b'. This corresponds to the inequality a ≤ x ≤ b. Example: [-3, 1] represents all numbers between -3 and 1, including -3 and 1.

• Half-Open Intervals: These combine open and closed intervals.

  • [a, b): Represents all values between 'a' and 'b', including 'a' but excluding 'b'. This corresponds to the inequality a ≤ x < b. Example: [0, 10) represents all numbers from 0 to 10, including 0 but excluding 10.

  • (a, b]: Represents all values between 'a' and 'b', excluding 'a' but including 'b'. This corresponds to the inequality a < x ≤ b. Example: (-5, 0] represents all numbers from -5 to 0, excluding -5 but including 0.

2. Unbounded Intervals

These intervals extend infinitely in one or both directions.

• (a, ∞): Represents all values greater than 'a'. This corresponds to the inequality x > a. Example: (3, ∞) represents all numbers greater than 3.

• [-∞, a): Represents all values less than or equal to 'a'. This corresponds to the inequality x ≤ a. Example: (-∞, -1] represents all numbers less than or equal to -1.

• (-∞, ∞): Represents all real numbers. This is equivalent to the statement -∞ < x < ∞.

3. Compound Inequalities

These inequalities involve multiple conditions. For example: -2 ≤ x < 5 means x is greater than or equal to -2 and less than 5. In interval notation, this is written as [-2, 5).

Step-by-Step Examples: Converting Inequalities to Interval Notation

Let's work through several examples to solidify your understanding:

Example 1: Write the inequality x ≥ 7 in interval notation.

Since x is greater than or equal to 7, the interval starts at 7 and extends to infinity. The bracket [ is used because 7 is included. The infinity symbol is always accompanied by a parenthesis.

Interval Notation: [7, ∞)

Example 2: Write the inequality -3 < x < 10 in interval notation.

This inequality states that x is greater than -3 and less than 10. Since both -3 and 10 are excluded, we use parentheses.

Interval Notation: (-3, 10)

Example 3: Write the inequality x ≤ -5 in interval notation.

This inequality indicates that x is less than or equal to -5. The interval extends from negative infinity up to and including -5.

Interval Notation: (-∞, -5]

Example 4: Write the inequality x > 2 or x < -1 in interval notation.

This is a compound inequality involving two separate conditions. We represent this using two separate intervals:

Interval Notation: (-∞, -1) U (2, ∞) The symbol "U" represents the union of the two sets.

Example 5: Solve the inequality 2x + 5 ≤ 11 and express the solution in interval notation.

1. Subtract 5 from both sides: 2x ≤ 6
2. Divide both sides by 2: x ≤ 3

Interval Notation: (-∞, 3]

Example 6: Solve the inequality |x - 3| < 2 and express the solution in interval notation.

This involves an absolute value inequality. We can rewrite this as:

-2 < x - 3 < 2

1. Add 3 to all parts of the inequality: 1 < x < 5

Interval Notation: (1, 5)

Example 7: Solve the quadratic inequality x² - 4x - 5 > 0 and express the solution in interval notation.

1. Factor the quadratic: (x - 5)(x + 1) > 0
2. Find the roots: x = 5 and x = -1
3. Test intervals: The parabola opens upwards, so the inequality is satisfied when x < -1 or x > 5.

Interval Notation: (-∞, -1) U (5, ∞)

Advanced Applications and Considerations

Interval notation is not just limited to simple inequalities. It plays a significant role in more advanced mathematical concepts:

• Calculus: Interval notation is crucial when defining domains and ranges of functions, finding intervals of increase or decrease, and identifying intervals of concavity.

• Real Analysis: Interval notation helps express solution sets of inequalities involving limits, derivatives, and integrals.

• Linear Programming: Interval notation is used to define feasible regions and optimize solutions.

Conclusion

Mastering interval notation is essential for anyone working with inequalities in mathematics. Its concise and standardized format makes it invaluable for representing solution sets, simplifying complex expressions, and facilitating communication in mathematical discussions. By understanding the different types of intervals and following the step-by-step examples provided, you can effectively convert inequalities into interval notation, empowering you to tackle more advanced mathematical concepts with confidence. Remember to practice regularly to reinforce your understanding and to become proficient in this crucial mathematical skill. The more you practice, the more intuitive and efficient this process will become.