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Chapter 0: Problem 4

Draw the following intervals on the number line. $$\left[1, \frac{3}{2}\right]$$

Short Answer

Expert verified
Draw a solid line from 1 to 1.5, with solid dots at both ends.

Step by step solution

01

- Understand the Interval

The interval \(\big[1, \frac{3}{2}\big]\) includes all the numbers between 1 and \(\frac{3}{2}\), including the endpoints 1 and \(\frac{3}{2}\).
02

- Identify the Endpoints

Identify the two endpoints of the interval: the number 1 and the number \(\frac{3}{2}\), which is equal to 1.5.
03

- Draw the Number Line

Draw a horizontal number line and mark the points 1 and 1.5 clearly on the line.
04

- Mark the Interval

Draw a solid dot at each of the endpoints, 1 and 1.5, because the interval includes these endpoints. Then draw a line segment connecting the dots from 1 to 1.5.

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Key Concepts

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

Number Line
A number line is a straight, horizontal line that helps visually represent numbers. Numbers assigned to points on this line progress from left to right. Negative numbers are placed to the left of zero, and positive numbers are to the right. Number lines are very useful in mathematics for comparing and understanding numbers more effectively. They serve as a handy tool to visually depict intervals, quick comparisons, and even solve equations. By marking numbers and segments on a number line, students can easily visualize concepts such as intervals and quantities.
Endpoints
Endpoints are the specific points that define the boundaries of an interval. For example, in the interval \(\big[1, \frac{3}{2}\big]\), the endpoints are 1 and 1.5. Endpoints can be included in the interval, which means the interval is closed, or they can be excluded, indicating an open interval. To include an endpoint, we use a square bracket [ ]. For excluding an endpoint, a parenthesis ( ) is used. In the given exercise, both endpoints are included, as denoted by the square brackets.
Closed Interval
A closed interval includes its endpoints. This means that every number between and including the endpoints is part of the interval. In interval notation, closed intervals are written with square brackets. For example, \(\big[1, \frac{3}{2}\big]\) is a closed interval, which includes both 1 and 1.5. This distinction is important for accurately identifying the range of numbers in the interval and ensuring clarity in mathematical communication. When drawing this on a number line, we use solid dots at the endpoints to indicate that those points are included.
Drawing Intervals
To draw an interval on a number line, follow these steps:
  • Draw a horizontal line to represent the number line.
  • Identify and mark the endpoints clearly. For the interval \(\big[1, \frac{3}{2}\big]\), mark the points 1 and 1.5 on the line.
  • Since the interval is closed, place a solid dot on each of the endpoints. This signifies that the endpoints are included in the interval.
  • Lastly, draw a line segment connecting the dots from 1 to 1.5. This represents all the numbers between 1 and 1.5.
Following these steps ensures the interval is accurately depicted, making it easier to understand.

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