Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. \((-\infty, 7]\)

Step-by-Step Solution

Answer

The inequality representation of the interval \((-\infty, 7]\) is \(x \leq 7\). The interval is unbounded as it does not have a lower limit.

Step 1: Express the interval as an inequality

The interval \((-\infty, 7]\) includes all real numbers from negative infinity to 7, including 7. The inequality that represents this interval is \(x \leq 7\) where x represents any real number.

Step 2: Determine if the interval is bounded or unbounded

An interval is bounded if it has both upper and lower limits. An interval is unbounded if it does not have either a lower limit, an upper limit, or both. Looking at the interval \((-\infty, 7]\), the lower limit is negative infinity and the upper limit is 7. Since infinity (whether negative or positive) is not a real number and it's not considered a 'bound', the interval \((-\infty, 7]\) is unbounded.

Key Concepts

Inequalities

Understanding inequalities is crucial when working with intervals on the number line. Inequalities describe the relationship between two expressions, showing if one is larger, smaller, or equal to the other.

For example, in the inequality \(x \leq 7\), it indicates that the variable \(x\) can be any number less than or equal to 7. Solving inequalities involves finding the range of values that satisfy this condition.

When dealing with inequalities, remember these comparisons:

• \( < \): less than
• \( \leq \): less than or equal to
• \( > \): greater than
• \( \geq \): greater than or equal to

This understanding helps to translate intervals into inequalities easily, especially when analyzing limits of a set or sequence.

Bounded and Unbounded Intervals

Intervals can be classified as either bounded or unbounded based on their limits. A bounded interval has finite upper and lower limits, while an unbounded interval extends indefinitely in at least one direction.

Consider the interval \((-\infty, 7]\). This interval is unbounded because it extends indefinitely towards negative infinity, with no lower limit. However, it does have an upper limit at 7, which is included in the interval (denoted by the closing bracket).

Types of intervals include:

• Bounded intervals: Both ends have finite limits, such as \([a, b]\).
• Unbounded intervals: One or both ends extend to infinity, such as \(a, \infty)\) or \((-\infty, b]\).

This classification helps determine the properties and behaviors of the interval in mathematical analysis.

Real Numbers

Real numbers include all the numbers on the continuous number line. This set encompasses rational numbers (fractions and integers) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). Real numbers can be positive, negative, or zero.

They are crucial when discussing intervals and inequalities, as these concepts rely on the complete set of real numbers to define the range of values. In interval notation, we specify parts of the real number line that meet certain conditions, using inequalities.

The interval \((-\infty, 7]\) uses real numbers less than or equal to 7. Even though infinity itself isn't a real number, the interval considers all negative real numbers extending towards it.

Understanding real numbers allows you to apply inequalities and interval analysis effectively, providing clarity in mathematical expressions and solutions.