Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. \([-1,5]\)

Step-by-Step Solution

Answer

The inequality that represents the interval \([-1,5]\) is -1 \(\leq\) x \(\leq\) 5, and the interval is bounded.

Step 1: Writing The Inequality

An inequality is written from a given interval by including the boundaries and using less than or equal to, greater than or equal to signs. When we look at the interval \([-1,5]\), it means all numbers from -1 to 5 inclusive. Thus, the inequality that represents the interval is -1 \(\leq\) x \(\leq\) 5. The variable 'x' represents all possible numbers in the interval.

Step 2: Determining If The Interval Is Bounded Or Unbounded

An interval is said to be bounded if it has both upper and lower limits. From the inequality -1 \(\leq\) x \(\leq\) 5, it can be seen that the number 'x' is bounded by -1 from below and 5 from above. Hence, the interval is bounded.

Key Concepts

Writing Inequalities

Understanding how to write inequalities is fundamental in math, as they are used to express relationships between numbers or expressions. When a range of numbers is defined, such as in the given interval \( [-1,5] \), it specifies that the values it includes are those from \( -1 \) to \( 5 \), and this range can be captured in the language of inequalities. With this concept, we translate the inclusive interval into an inequality. The symbols \( \leq \) (less than or equal to) and \( \geq \) (greater than or equal to) are key for representing that the boundaries are part of the set. Here, we use the inequality \( -1 \leq x \leq 5 \) to denote that \( x \) can be as small as \( -1 \) and as large as \( 5 \), including both endpoints in the set of possible numbers.

Bounded Intervals

A bounded interval is like a closed corridor with clearly marked ends; it contains a defined set of numbers lying between a lower and an upper limit. In the context of inequalities and interval representation, when we have both endpoints specified, like in \( [-1,5] \) from our exercise, we are dealing with a bounded interval. This means that the values of \( x \) are restricted and cannot go beyond the limits of \( -1 \) and \( 5 \). To determine if an interval is bounded, we look for both the smallest and largest numbers that are included. If these exist and are finite, we indeed have a bounded interval. In contrast, intervals that stretch indefinitely in one or both directions (such as \( (-\infty,3) \) or \( (4,\infty) \) ) are known as unbounded intervals.

Interval Notation

Interval notation is a shorthand used to describe the set of numbers that satisfy an inequality. It's a clear and concise way to communicate vast ranges without writing long and complex inequalities. In our example, \( [-1,5] \) uses square brackets to indicate that the interval includes its endpoints \( -1 \) and \( 5 \). This sort of notation allows anyone reading to instantly understand the boundaries of an interval. Different symbols are used to denote if the ends are included (closed interval with square brackets like \( [ \) and \( ] \) ) or not included (open interval with parentheses like \( ( \) and \( ) \) ). Understanding this notation is key to grasping the extent of a given interval quickly and accurately.