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Chapter 0: Problem 5

Draw the following intervals on the number line. $$ (-\infty, 3) $$

Short Answer

Expert verified
Shade the number line to the left of 3 with an open interval symbol (parenthesis) at 3.

Step by step solution

01

- Identify the Interval Type

The interval \((-\infty, 3)\) starts from negative infinity and goes up to 3, excluding 3. This is an open interval on the right side.
02

- Understand Negative Infinity

Negative infinity (\(-\infty\)) represents all numbers less than a certain value and extends indefinitely. Always remember, negative infinity is not a number you can reach or draw; it is a concept of extending infinitely in the negative direction.
03

- Write Down the Endpoint

Identify and write down the endpoint of the interval, which is the number 3 in this case. Since this is an open interval, 3 will not be included. This means you will use a parenthesis (rather than a bracket) at 3.
04

- Draw the Number Line

Draw a horizontal line and mark it with numbers. Ensure the number 3 is clearly indicated.
05

- Shade the Interval

Shade the number line to the left of 3, indicating that all numbers less than 3 are included in the interval. Use a parenthesis at 3 to show that 3 is not included in the interval.

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Key Concepts

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

Understanding the Number Line
A number line is a straight, horizontal line used to represent numbers. It helps visualize concepts in math, such as intervals, positioning, and relationship between numbers.
Numbers on a number line usually increase from left to right. For example, negative values are placed on the left side of zero (0), and positive numbers are on the right.

To draw an interval on a number line:
  • Identify the relevant endpoints and their types (open or closed).
  • Mark the endpoints accordingly.
  • Shade the appropriate section to indicate the range of the interval.
Introduction to Open Intervals
Open intervals are a way to describe a range of numbers without including the endpoints. In mathematical notation, an open interval is written using parentheses. For example, \(a, b\) means that the interval starts just after point \(a\) and goes up to, but does not include, point \(b\).

In the problem \(\(-\infty, 3\)\), the interval starts from negative infinity and goes up to 3, excluding the number 3 itself.
When representing an open interval on a number line:
  • Use a parenthesis to mark the endpoint that's not included.
  • Shade the line towards the direction that includes the valid range (left for negatives, right for positives).
Negative Infinity Explained
Negative infinity (\(-\infty \)) is a concept, not an actual number, that represents an infinite extension in the negative direction. It suggests that the numbers keep getting smaller and smaller endlessly.

When dealing with intervals involving negative infinity, you are essentially considering all numbers that are less than a given point. You cannot mark negative infinity on the number line because it extends indefinitely to the left, but you understand it conceptually.
  • Use an open-ended approach on a number line to represent extending towards negative infinity.
  • Remember, there is no lower limit when considering negative infinity.

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Most popular questions from this chapter

Velocity When a car is moving at \(x\) miles per hour and the driver decides to slam on the brakes, the car will travel \(x+\frac{1}{20} x^{2}\) feet. (The general formula is \(f(x)=a x+b x^{2}\), where the constant \(a\) depends on the driver's reaction time and the constant \(b\) depends on the weight of the car and the type of tires.) If a car travels 175 feet after the driver decides to stop, how fast was the car moving? (Source: Applying Mathematics: A Course in Mathematical Modelling.)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\frac{x^{3}}{y^{-2}}\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\frac{-x^{3} y}{-x y}\)

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}}\). Calculate the following functions. Take \(x>0\). \([f(x) g(x)]^{3}\)

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}}\). Calculate the following functions. Take \(x>0\). \(\sqrt[3]{f(x) g(x)}\)
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