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Chapter 0: Problem 1

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

Short Answer

Expert verified
In the set-builder notation, the given interval is represented as \( S = \{x \in \mathbb{R} | 1 < x \le 6\} \). And in the number line, we illustrate it with starting from an open circle at 1 and ending at a fully filled circle at 6.

Step by step solution

01

Understand the Notation of Interval

The notation (1,6] represents an interval that includes all real numbers between 1 and 6. The round bracket ( ) means that the number next to it is not included in the interval, while the square bracket [ ] means that the number next to it is included. This means we have all the numbers greater than 1 and less than or equal to 6.
02

Set-builder Notation

The set-builder notation for this interval can be represented as \(S = \{x \in \mathbb{R} | 1 < x \leq 6\}\). This says that we are constructing a set S of all real numbers x such that x is greater than 1 and less than or equal to 6.
03

Graphing on a Number Line

Start by drawing a straight number line. Mark points 1 and 6 on the number line. Since the number 1 is not inclusive, we use an open circle at the point 1. Because 6 is inclusive, we use a filled circle at 6. Draw a line between the circle at 1 and the circle at 6 to represent all the numbers in between.

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Most popular questions from this chapter

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{2}+36=(x+6)^{2}$$

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).

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