Problem 14 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 3.5]$$

Step-by-Step Solution

Answer

The set-builder notation for the interval \( (-\infty, 3.5] \) is \( \{ x : x \in \mathbb{R}, x \leq 3.5 \} \). A graph of this interval on a number line would include a solid circle at 3.5, with an arrow pointing leftwards, towards negative infinity.

Step 1: Understanding the Interval Notation

Consider the interval given as \( (-\infty, 3.5] \). The parenthesis '(' denotes that the interval starts from negative infinity and does not include any real number i.e., the interval is open towards negative infinity. The square bracket ']' indicates that the interval ends at 3.5 and includes the number 3.5 i.e., the interval is closed at 3.5.

Step 2: Converting to Set-builder Notation

Set-builder notation provides a precise method to describe sets. The interval notation can be translated into set builder notation as: \( S = \{ x : x \in \mathbb{R}, x \leq 3.5 \} \). Meaning, S is the set of all x such that x is a real number and x is less than or equal to 3.5.

Step 3: Graphing the Interval on a Number Line

To graph the interval on a number line, start with a straight horizontal line. This represents the number line. Draw a filled circle at the point corresponding to the number 3.5, as it is included in the interval. Draw an arrow pointing to the left starting from the filled circle to represent all numbers less than 3.5. The arrow continues indefinitely to represent the negative infinity.