Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. \((11, \infty)\)

Step-by-Step Solution

Answer

The inequality that represents the interval \( (11, \infty) \) is 'x > 11' and the interval is unbounded.

Step 1: Translate Interval Notation to Inequality Notation

The given interval is \( (11, \infty) \). In interval notation, '()' indicates 'less than' or 'greater than' but not 'equal to'. So, this translates into the inequality 'x > 11'.

Step 2: Determine if the Interval is Bounded or Unbounded

An interval is bounded if it has both a lower and an upper bound. As the provided interval is \( (11, \infty) \), it has a lower bound (11) but no upper bound, as the value can go to infinity. This makes the interval unbounded.

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare two quantities. They tell us how one value relates to another, whether it’s greater than, less than, or neither.
Understanding inequalities requires familiarizing yourself with these common symbols:

• > : Greater than
• < : Less than
• ≥ : Greater than or equal to
• ≤ : Less than or equal to

Inequalities can be open or closed. Open inequalities use the symbols '<' or '>', suggesting that the value is not included. For example, in the inequality \( x > 11 \), the number 11 is not part of the solution—a common aspect of half-open intervals where one limit is specified but not included.
When solving or graphing inequalities, always keep in mind to perform equivalent operations to maintain their truth, and remember to reverse the inequality symbol when multiplying or dividing by a negative number.

Bounded and Unbounded Intervals

Intervals in mathematics describe the range of numbers between two endpoints. These intervals can be either bounded or unbounded, which helps define what numbers belong to the interval.

• Bounded Intervals: These have specific starting and ending points. Both endpoints are finite numbers. For example, \([2, 8]\) encloses all numbers between 2 and 8, including these numbers themselves if square brackets are used.
• Unbounded Intervals: These have at least one endpoint that extends to infinity. For example, the interval \((11, \infty)\) begins at 11 and continues indefinitely. The number 11 serves as a boundary point, but the interval stretches infinitely beyond in one direction.

Recognizing whether an interval is bounded or unbounded is essential. It tells you about the limits of possibilities within a problem, such as understanding if there's a cap or limit or if values can continually increase or decrease without end.

Interval Notation Translation

Interval notation condenses the expression of a set of numbers on a number line. It's similar to inequalities yet more compact.
Interval notation uses brackets and parentheses to identify the endpoints of intervals:

• '()' : Indicates that the endpoint is not included, aligning with the inequalities '<' or '>'.
• '[]' : Means that the endpoint is included, representing '≤' or '≥'.

To translate interval notation to an inequality, observe the symbols and adjust the expression accordingly. For instance, \((11, \infty)\) translates to \(x > 11\) because the interval starts right after 11 and extends indefinitely. There is no upper bound, reflecting the infinite aspect with 'infinity'.Understanding this translation not only ties into solving math problems but also helps when interpreting graphs and real-life situations where constraints need clear representation.