Problem 1 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$(1,6]$$

Step-by-Step Solution

Answer

The set-builder notation for the interval $(1,6]$ is \( x | 1 < x \leq 6 \). On a number line, this set is represented by a shaded or marked area between an open circle on 1 (excluding 1) and closed circle on 6 (including 6).

Step 1: Conversion to Set-builder Notation

To express the provided interval $(1,6]$ in set-builder notation, a format is used that describes the set containing all numbers on the real line that falls between 1 and 6, excluding 1 but including 6. The notation becomes \( x | 1 < x \leq 6 \). This reads as 'The set of all x such that x is greater than 1 and less than or equal to 6'.

Step 2: Graphing on a Number Line

To graph this interval on a number line, one begins by drawing a straight line, labeling it with numbers from 1 to 6. Since 1 is excluded from the set, an open circle is placed just above number 1 on the line to recognize this. Because 6 is included in the set, a closed or filled circle is placed just above number 6. The area between the open circle on 1 and the closed circle on 6 is shaded or marked off to show all the values within this range.