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Chapter 2: Problem 9

Write each inequality in interval notation, and graph the interval. \(k \leq 4\)

Short Answer

Expert verified
Interval: \((-\infty, 4]\). Closed circle at 4.

Step by step solution

01

Understand the Inequality

The inequality given is \(k \leq 4\). This means that \(k\) can be any number that is less than or equal to 4.
02

Write in Interval Notation

To represent \(k \leq 4\) in interval notation, note that \(k\) starts from negative infinity and goes up to 4, including 4. Therefore, the interval notation is \( (-\infty, 4] \).
03

Graph the Interval

On the number line, draw a line that extends from negative infinity up to and including the number 4. Use a closed circle at 4 to indicate that 4 is included in the interval.

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Key Concepts

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

inequality notation
Inequality notation is a way to express the range of values that a variable can take. Unlike an equation, which shows equality, an inequality shows that one side can be larger or smaller than the other. In the context of our exercise, we have the inequality: \(k \leq 4\). This tells us that the variable \(k\) can be any value that is less than or equal to 4. Here are some common inequality symbols you may encounter:
  • \(<\): less than
  • \(>\): greater than
  • \(\leq\): less than or equal to
  • \(\geq\): greater than or equal to
Understanding these symbols is critical to interpreting and solving inequalities correctly. When you see \(k \leq 4\), it means we include the number 4, which is why we use a closed circle when graphing.
interval notation
Interval notation provides a compact way to describe an interval of values on the number line. It uses parentheses and brackets to indicate the endpoints of the interval. Let's break it down using our example \(k \leq 4\):
  • **Parentheses \(()\)**: Used when the endpoint is not included in the interval.
  • **Brackets \([]\)**: Used when the endpoint is included in the interval.
So, the interval notation for \(k \leq 4\) would be \((-\infty, 4]\). Here:
  • **\(-\infty\)**: Represents all numbers extending infinitely to the left.
  • **\(4\)**: This is our endpoint, and we include 4, shown by the bracket \([\).
Interval notation makes it simpler to write and understand intervals, especially when working with inequalities.
number line graphing
Graphing inequalities on a number line helps visualize the set of solutions. Here's a simple guide to graphing \(k \leq 4\) on a number line:
  • **Step 1**: Draw a horizontal line. This will be our number line ranging from negative to positive infinity.
  • **Step 2**: Identify the critical point on the number line (in this case, 4).
  • **Step 3**: Since 4 is included in the set of solutions (because of the \(\leq\) sign), draw a closed circle or dot at 4.
  • **Step 4**: Shade or draw an arrow extending to the left from 4 to indicate all numbers less than 4 are included.
This visual representation helps to easily identify which values are included within the inequality. For inequalities that do not include the endpoint, you would use an open circle instead. Graphing inequalities on a number line solidifies understanding and aids in pattern recognition.

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