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Chapter 2: Problem 4

Work each problem. How does the graph of \(x \geq-7\) differ from the graph of \(x>-7 ?\)

Short Answer

Expert verified
The graph of \(x \, \geq \, -7\) includes \(-7\) (solid dot), while the graph of \(x > -7\) does not include \(-7\) (open circle).

Step by step solution

01

Understand the inequality

Consider the inequality: \(x \, \geq \, -7\). This means that \(x\) can be any number greater than or equal to \(-7\).
02

Graph the inequality \(x \, \geq \, -7\)

On a number line, \(x \, \geq \, -7\) includes all points starting from \(-7\) and moving to the right. Represent \(-7\) with a solid dot to show that \(-7\) is included in the solution.
03

Understand the second inequality

Consider the inequality: \(x > -7\). This means that \(x\) can be any number strictly greater than \(-7\).
04

Graph the inequality \(x > -7\)

On a number line, \(x > -7\) includes all points starting to the right of \(-7\). Represent \(-7\) with an open circle to show that \(-7\) is not included in the solution.
05

Compare the graphs

The primary difference between the graphs of \(x \, \geq \, -7\) and \(x > -7\) is at the point \(-7\). In the graph of \(x \, \geq \, -7\), \(-7\) is included and represented with a solid dot. In the graph of \(x > -7\), \(-7\) is not included and represented with an open circle.

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Key Concepts

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

inequality representation
Inequalities are mathematical expressions that show the relationship between two values where one is either less than, greater than, or equal to the other. In the exercise, we explored two types of inequalities: \(x \geq -7\) and \(x > -7\). For \(x \geq -7\), the symbol \(\geq\) means greater than or equal to. This indicates that \(x\) can be \textbf{any value equal to or more than -7}. For \(x > -7\), the symbol \(>\) means strictly greater than. This shows that \(x\) can be \textbf{any value more than -7}, but not -7 itself. Understanding the difference between these symbols is essential as it affects how we graph these inequalities and what values are considered solutions.
number line graphing
Graphing inequalities on a number line is a useful method to visually represent the range of values that satisfy the inequality. To graph \(x \geq -7\), we:
  • Start at \textbf{-7} on the number line.
  • Use a solid dot at \textbf{-7} to indicate this point is included.
  • Draw a line extending to the right to show all numbers greater than \textbf{-7} are included.
For \(x > -7\), the process is slightly different:
  • Begin at \textbf{-7} on the number line.
  • Use an open circle at \textbf{-7} to show this point is not included.
  • Extend the line to the right to represent all numbers greater than \textbf{-7} without including \textbf{-7} itself.
Visual representation on the number line helps to easily understand and compare different inequalities.
solid dot vs open circle
When graphing inequalities on a number line, the use of a solid dot or an open circle is key to correctly interpreting the solution set. A solid dot is used to:
  • Represent that a particular number is included in the solution.
  • For example, in \(x \geq -7\), the number \textbf{-7} is part of the solutions, and thus depicted with a solid dot.
On the other hand, an open circle is used to:
  • Show that a particular number is not included in the solution set.
  • For example, in \(x > -7\), the number \textbf{-7} is not part of the solutions and therefore is represented with an open circle.
This simple distinction helps to avoid confusion and accurately represents whether a boundary value is part of the solution.

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