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Chapter 6: Problem 2

Graph each set of real numbers on a number line. \(\\{x \mid x>-2\\}\)

Short Answer

Expert verified
The number line consists of an open circle at -2 and an arrow extending towards positive infinity from that point, which represents all real numbers greater than -2.

Step by step solution

01

Understand the number set

The number set here is \(x \mid x>-2\). It represents all numbers greater than -2. Therefore, the aim is to plot all real numbers greater than -2 on a number line.
02

Plot -2 on the line

First, draw a horizontal line to represent the number line. Mark a point at -2. Since the exercise specifies values that are strictly greater than -2, the point at -2 should be represented with an open circle or parenthesis, indicating -2 itself is not included.
03

Represent numbers greater than -2

Draw an arrow to the right of the point marked -2, that extends indefinitely. This arrow represents all real numbers greater than -2.

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Key Concepts

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

Number Line
The number line is a visual representation of numbers placed on a straight horizontal line. It helps us to understand the size and order of our numbers. When graphing inequalities, the number line becomes a handy tool.
Each point on the line corresponds to a numerical value. These values increase as we move from left to right.
  • In the context of the problem, by drawing an arrow on the line, we illustrate which direction our solution extends.
  • Understanding how to accurately position and interpret numbers on this line is crucial for graphing inequalities correctly.
The starting point often uses markers, like circles or lines, to show whether the end point is included in the inequality representation.
Real Numbers
Real numbers are essentially all the numbers you can think of. They encompass whole numbers, fractions, decimals, and even irrational numbers like \(\pi\)or the square root of 2.
Whether positive, negative, or zero, any number that can be plotted on a number line is considered part of the real numbers.
  • This broad category is what we refer to when specifying solutions to inequalities involving x, such as in the given exercise.
  • In inequalities, real numbers help in identifying the entire range of possible solutions.
When graphing a set of real numbers on a number line, we're indicating every potential value in our solution set.
Open Circle
An open circle on a number line is used to indicate that a particular value is not included in the set of solutions.
In the exercise, -2 is represented with an open circle because x must be greater than -2.
  • This is different from a closed circle, which would represent that the number itself is part of the solution set.
  • Using an open circle effectively communicates that the boundary value is not included.
This visual cue is essential in making sure everyone understands the exact set of numbers being represented.
Greater Than Symbol
The greater than symbol \(>\) is crucial in understanding inequalities. It tells us which values of x are valid solutions to the inequality.
In the context of the problem, the symbol \(x > -2\) directs us to consider all numbers larger than -2.
  • It ensures clarity by pointing out that the number immediately to the right of -2 and onward is part of our solution.
  • The symbol naturally suggests a directionality, indicating that we should shade or draw an arrow to the right on the number line.
Recognizing and interpreting the greater than symbol is key to accurately graphing and solving inequalities on number lines.

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