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Chapter 2: Problem 14

Graph each inequality, and write it using interval notation. \(m>5\)

Short Answer

Expert verified
Graph with an open circle at 5 and shade to the right. Interval: \((5, \infty)\).

Step by step solution

01

- Identify the Inequality

Recognize that the given inequality is in the form of a linear inequality. Here, we have: \( m > 5 \)
02

- Determine the Boundary Point

Identify the boundary point from the inequality. The boundary point is 5. Note that this point is not included in the solution since it's a strict inequality (greater than, not greater than or equal to).
03

- Graph the Inequality

To graph the inequality: - Draw a number line.- Place an open circle at 5 to show that 5 is not included in the solution.- Shade the number line to the right of 5 to represent all values greater than 5.
04

- Write in Interval Notation

Translate the inequality \( m > 5 \) into interval notation. Since 5 is not included, use a parenthesis: \( (5, \, \text{infinity}) \).

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Key Concepts

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

Linear Inequalities
A linear inequality in one variable is similar to a linear equation but instead of the equality sign, it uses inequality symbols such as >, <, 鈮�, or 鈮�. This means the set of solutions includes a range of values rather than a single number.
In the example given, the inequality is
\( m > 5 \). This indicates that \( m \) can be any number greater than 5. Linear inequalities are essential in representing ranges rather than exact values.
To solve linear inequalities, you follow a process similar to solving linear equations, but always remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
Boundary Point
The boundary point is a crucial concept in graphing inequalities because it defines the limit where the inequality starts or ends. In our example, the boundary point is 5.
Since our inequality is \( m>5 \), we place an open circle at 5 on the number line to indicate that 5 is not part of the solution.
This is different from 鈮� or 鈮� inequalities, where you would use a closed circle showing the boundary point is included in the solution.
Understanding the boundary point helps ensure accurate representation and interpretation of the inequality on a number line.
Interval Notation
Interval notation is a way of writing subsets of the real number line and is often used to express the solution set of inequalities. It offers a concise method to describe the range of possible values.
For \( m > 5 \), we use the symbol \(( \) to denote that 5 is not included in the interval, and \(\text{infinity}\) indicates the values continue indefinitely. Therefore, the interval notation is written as \((5, \, \text{infinity})\).
Some key points to remember include:
  • Use \([ \) or \( ] \) for closed intervals where the endpoints are included.
  • Use \(( \) or \( ) \) for open intervals where the endpoints are not included.
  • \(\text{Infinity} \) and \(-\text{infinity} \) are always written with parentheses because they are not specific numbers that can be included in the interval.

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