Problem 43 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$[1,4]$$

Step-by-Step Solution

Answer

Inequality: \(1 \leq x \leq 4\). Number line: solid dots at 1 and 4, shaded in between.

Step 1: Understand the Interval Notation

The interval notation \([1, 4]\) represents all the numbers between 1 and 4, including the endpoints 1 and 4 themselves. The square brackets indicate that both endpoints are included in the interval.

Step 2: Convert to Inequality Notation

Convert the interval \([1, 4]\) into inequality notation. Since 1 and 4 are included in the interval, the inequality will include 'less than or equal to' signs: \(1 \leq x \leq 4\). This means \(x\) is a number that is greater than or equal to 1 and less than or equal to 4.

Step 3: Draw the Number Line

To represent the interval \([1, 4]\) on a number line, draw a line and mark the numbers 1 and 4 on it. Since both are included in the interval, draw solid dots at each of these points to indicate inclusion. Then, shade the line segment between these two dots, representing all numbers in the interval from 1 to 4, inclusive.

Key Concepts

Interval Notation

Interval notation is a method used to describe a range of numbers. It combines two numbers with brackets to show which numbers the interval includes or excludes. For example, the interval \([1, 4]\) includes all numbers from 1 to 4, where both endpoints are included.

• Square brackets \([\text{ and } ]\) mean the endpoint is included, like \([1, 4]\).
• Round brackets \((\text{ and } )\) mean the endpoint is not included, like \((1, 4)\).

Understanding interval notation is essential when expressing solutions to inequalities. By using it, you can clearly communicate whether the numbers at the boundary are part of the solution.

Number Line

A number line is a visual tool that helps you see the relationships between numbers. It's like a ruler where each point corresponds to a number. To represent \([1, 4]\) on a number line, you draw a line, mark the numbers 1 and 4, and use solid dots to indicate those numbers are part of the interval.

• Place solid dots on the number line at points that are included in the interval.
• Shade or color the line segment between the two dots if all numbers in between are part of the interval.

This visual representation makes it straightforward to understand which numbers are part of the solution to an inequality.

Inequality Representation

Inequality representation is a way to express the set of numbers that satisfy a given condition or equation. For the interval \([1, 4]\), it's represented as \(1 \leq x \leq 4\). This reads as "\(x\) is greater than or equal to 1 and less than or equal to 4."

• \(\leq\) and \(\geq\) symbols are used to express "less than or equal to" and "greater than or equal to."
• Simple inequalities without the equals line (\(<\) and \(>\)) are used for exclusive boundaries.

Inequality notation is crucial for solving real-world problems where ranges of values are considered. It's a concise way to indicate which solutions fit the given criteria.