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Chapter 1: Problem 48

Sketch the given set on a number line. $$\\{x |-1

Short Answer

Expert verified
Open circle at -1, shaded region to 3 with a closed circle at 3.

Step by step solution

01

- Understand the Set Notation

The set notation \(\{x | -1 < x \leq 3\}\) contains all real numbers \(x\) that are greater than -1 and less than or equal to 3.
02

- Identify the Boundaries

The boundaries for the set are -1 and 3. The inequality \(-1 < x\) means that -1 is not included in the set, while the inequality \(x \leq 3\) indicates that 3 is included in the set.
03

- Draw the Number Line

Draw a horizontal line and mark points -1 and 3 on it. This will serve as the reference for placing the relevant symbols.
04

- Plot the Interval

Since -1 is not included, use an open circle at -1. Since 3 is included, use a closed circle at 3. Shade the region in between the two points to show all numbers between -1 and 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers are all the numbers on the number line, including both rational and irrational numbers. Rational numbers can be written as fractions, like 1/2 or -3/4, whereas irrational numbers cannot be written as fractions and have non-repeating, non-terminating decimals, like \(\frac{\pi}{2}\) or \(\frac{\sqrt{2}}{3}\). Real numbers also include integers and zero.

In the context of the given set notation \({x | -1 < x \leq 3}\), we consider all real numbers between -1 and 3 without including -1 but including 3. This means that all numbers, whether they are whole, fractions, or irrational, are part of the set if they fall within this range.
Inequalities
Inequalities are mathematical expressions that show the relation between two values. In our example, we have combined inequalities: \(-1 < x \leq 3\).

The inequality \(-1 < x\) means that all numbers greater than -1 are part of the set. Since this is a 'less than' sign without an 'equal to', -1 is not included.

On the other hand, \(\text{x} \leq 3\) means all numbers up to and including 3 are part of the set. The 'less than or equal to' sign includes the endpoint, which is 3 in this case.

Understanding these inequalities helps in correctly placing the boundaries while graphing intervals on a number line.
Graphing Intervals
To graph intervals, you use a number line and symbols to indicate which parts are included or not included in the set.

For the set \({x | -1 < x \leq 3}\), you will start by:
  • Drawing a horizontal line to represent the number line.
  • Marking the points -1 and 3 on the line as these are the boundary values.


- Since \(-1 \lt x\), place an open circle at -1 to show that -1 itself is not part of the set.

- Since \(\text{x} \leq 3\), place a closed circle at 3 to include 3 in the set.



Once the endpoints are marked, shade the entire region between -1 and 3. This visually represents all the numbers that satisfy the inequality.

Graphing in this manner helps in easily visualizing the set and the range of values included in it.

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Most popular questions from this chapter

Graph the given set on a number line. $$\\{0,3,7\\}$$

Compute the given expression. Round off your answer to two decimal places where necessary. $$\frac{-8.4}{1.2}$$

Evaluate the given expression. Remember to follow the order of operations. $$6+4(3+2)$$

In Exercises \(73-84\), evaluate the given expression. $$7-3(5-1)$$

Locate the given number between two successive integers on the number line. For example, if the given number is \(2.6,\) it is located between 2 and 3 on the number line. $$\frac{-15}{2}$$
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