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Chapter 2: Problem 23

Write the solution set in interval notation and graph it. \(x>20\)

Short Answer

Expert verified
The solution set is \((20, \, +fty)\) and is represented on a number line with an open circle at 20 and shading to the right.

Step by step solution

01

Understand the Inequality

The inequality given is \(x > 20\). This means that we are looking for all the values of \(x\) that are greater than 20.
02

Write the Solution Set in Interval Notation

For the inequality \(x > 20\), the solution set includes all real numbers greater than 20 but not including 20. In interval notation, this is written as \((20, \, +fty)\).
03

Graph the Solution

To graph \(x > 20\), draw a number line. Place an open circle on the number 20 to indicate that 20 is not included in the solution set. Shade the line to the right of 20 to show that all numbers greater than 20 are included in the solution set.

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Key Concepts

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

solution set
The term 'solution set' refers to the collection of all possible values that satisfy a given inequality or equation. In the case of the inequality \(x > 20\), the solution set encompasses all values of \(x\) that are greater than 20. It鈥檚 important to understand that this does not include the number 20 itself, as the inequality specifies values strictly greater than 20.
To determine the solution set for any inequality, follow these steps:
  • Identify the type of inequality (greater than, less than, etc.).
  • Locate the boundary value (in this case, 20).
  • Determine which side of the boundary includes the solution (greater than or less than).
Once these steps are completed, you will have a clear picture of all the values that belong to the solution set.
interval notation
Interval notation is a way of writing subsets of the real number line. By using parentheses and brackets, we can compactly represent the continuous range of values that satisfy an inequality.
For the inequality \(x > 20\), the interval notation is written as \((20, \, \infty)\). Here鈥檚 a breakdown of what this means:
  • The parentheses \(()\) indicate that 20 is not included in the solution set (an open interval).
  • The infinity symbol \(\infty\) denotes that the set extends indefinitely to the right.
This notation is extremely useful because it provides a concise and precise way to convey the idea of all numbers greater than 20. Remember, parentheses signify values that are not included, while brackets \([]\) would denote values that are included in the range.
graphing inequalities
Graphing inequalities on a number line helps visualize the solution set. For \(x > 20\), graphing proceeds as follows:
  • First, draw a horizontal number line with appropriate markings.
  • Locate and mark the number 20 on the line.
  • Since 20 is not included in the solution, place an open circle (or hollow dot) on it.
  • Shade the line extending to the right of the open circle, indicating all numbers greater than 20 are part of the solution.
This visual representation makes it easier to understand which values satisfy the inequality. The open circle clearly shows that 20 itself is not part of the solution, while the shaded region depicts that all values beyond 20 are included.
For any inequality, always remember to:
  • Use open circles for \(>\) and \(<\) (strict inequalities).
  • Use closed circles for \(\geq\) and \(\leq\) (inclusive inequalities).
  • Consistently shade the number line in the direction that represents the correct set of solutions.

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Most popular questions from this chapter

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