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Chapter 1: Problem 6

Using interval notation, write each set. Then graph it on a number line. $$\\{x |-5

Short Answer

Expert verified
The interval notation is \((-5, -4]\) and is graphed with an open circle at \(-5\), closed circle at \(-4\), and the region in between shaded.

Step by step solution

01

Expression Analysis

Understand that the given set includes numbers greater than \(-5\) and up to and including \(-4\). This is represented as \(-5<x\leq-4\).
02

Convert to Interval Notation

Change the set builder notation to interval notation. Since the numbers are greater than \(-5\) but include \(-4\), the correct interval is \((-5, -4]\).
03

Graph on a Number Line

Draw a number line and indicate the numbers involved. Place an open circle at \(-5\) (since \(-5\) is not included) and a closed circle at \(-4\) (since \(-4\) is included). Shade the region between these two points to represent the set of numbers.

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Key Concepts

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

Set Builder Notation
Set builder notation is a compact way to define a set by specifying the properties that its members must satisfy. In our example, the set is written as \({x | -5
  • The variable (in this case, \(x\)) represents 'any' member of the set.
  • The bar \(|\) separates the variable from its condition.
  • The inequalities describe the relationship the variable must satisfy.
Number Line
A number line provides a visual representation of numbers in a linear order. It can effectively illustrate where a set begins and ends. Each point on the number line corresponds to a real number, which helps in understanding the relationships between these numbers. In the context of our example, a number line is used to graph the interval \((-5, -4]\).
  • An open circle or parenthesis indicates that a number is not part of the set. In our case, there is an open circle at \(-5\).
  • A closed circle or bracket means the number is included in the set. For the interval \((-5, -4]\), there is a closed circle at \(-4\).
  • The shaded region between \(-5\) and \(-4\) demonstrates all the potential values of \(x\) that satisfy the set's conditions.

Using number lines is a great way to "see" or "picture" how real numbers relate to each other, making abstract ideas more concrete.
Inequalities
Inequalities are mathematical expressions used to show how one value compares to another. They are crucial in describing intervals or sets. In the set \({x | -5
  • \(-5 < x\) means \(x\) must be greater than \(-5\), but not equal.
  • \(x \leq -4\) means \(x\) can be less than or equal to \(-4\).

When using inequalities, the symbols \(<\), \(>\), \(\leq\), and \(\geq\) help to specify exact constraints. Understanding these symbols is essential for solving and interpreting mathematical problems, as they denote if endpoints of an interval are included or excluded. With practice, you'll find inequalities offer a precise and simple means to define much about how numbers relate.

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Most popular questions from this chapter

Using the variable \(x\), write each interval using set-builder notation. $$(-4,3)$$

Find the equation of the line satisfying the given conditions, giving it in slope-intercept form if possible. Perpendicular to \(x=3,\) passing through \((1,2)\)

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c|c} \boldsymbol{x} & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} \\\ \hline \boldsymbol{y} & 0 & -1 & -2 & -3 & -4 \end{array}$$

Sketch the graph of \(f\) by hand. $$f(x)=-4$$

Using interval notation, write each set. Then graph it on a number line. $$\\{x | 1 \leq x<2\\}$$
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