Question

Graph the inequality. \(x \geq-6\)

Short answer

The graph of the inequality \(x \geq-6\) is a shaded portion on the number line including and to the right of -6. The point -6 is marked with a closed circle to indicate it is included in the solution set.

Step-by-step solution

Analyze the inequality

The inequality \(x \geq-6\) means that x takes any value greater than or equal to -6. It's important to note the difference between 'greater than' and 'greater than or equal to'. In this case, since it is 'greater than or equal to', the point -6 itself is included in the solution.

Plot a number line

Firstly, draw a horizontal line representing the number line. Mark a point for -6 on the number line.

Shade the number line

Since the inequality \(x \geq-6\) includes -6, mark the point -6 with a closed circle to indicate its presence in the solution set. Then, shade all numbers to the right of and including -6, to show that all these numbers are greater than or equal to -6.

Key concepts

Understanding the Number Line

A number line is a visual representation of numbers laid out in a straight line. It serves as a foundational tool in mathematics to illustrate numerical order and understand concepts such as counting, addition, subtraction, and more complex topics like inequalities. To graph inequalities effectively, we must first comprehend how a number line is set up.

Typically, a number line is drawn as a horizontal line with positive numbers extending to the right and negative numbers to the left of a central '0'. Each point on the line corresponds to a unique number, and the distance between any two points represents the numerical difference between those numbers.

To use a number line for graphing an inequality, such as in the exercise \(x \geq -6\), one begins by marking the boundary number, in this case, -6. From there, depending on the nature of the inequality, we determine the direction in which to shade the line, which visually represents all possible solutions to the inequality.

Inequality Notation Explained

Inequality notation is a compact way to express a range of numbers that satisfy a particular condition. Instead of listing out every single number, we use symbols to succinctly show how one quantity relates to another. These symbols include \(>\), \(\geq\), \(<\), and \(\leq\), which stand for 'greater than', 'greater than or equal to', 'less than', and 'less than or equal to', respectively.

In the exercise \(x \geq -6\), the notation \(\geq\) signals that the value of \(x\) is not just larger than -6, but can also be exactly -6. Hence, inequalities are about ranges of numbers where a solution can be more than one value, and it is crucial for students to understand and interpret these symbols correctly for graphing purposes.

Closed Circle Notation on Number Lines

When graphing inequalities on a number line, the use of 'open' and 'closed' circles plays an important role in accurately conveying the nature of the solutions. These circles are not just arbitrary marks; they communicate whether a boundary number is included or excluded from the solution set to an inequality.

A closed circle, which is filled in, illustrates that the boundary number is part of the solution set. This corresponds to inequalities with the 'greater than or equal to' (\geq) or 'less than or equal to' (\leq) symbol. In contrast, an open circle, which is an unfilled or hollow circle, denotes that the boundary number is not included in the solution and is used for strict inequalities represented by 'greater than' (>) or 'less than' (<) alone.

Thus, the closed circle at -6 in the given exercise \(x \geq -6\) is a crucial element. It visually conveys to anyone interpreting the graph that -6 is a valid solution to the inequality, in addition to all the numbers to its right on the number line.