/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Express each interval using ineq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 51

Express each interval using inequality notation and show the given interval on a number line. $$(-\infty, 1)$$

Short Answer

Expert verified
The interval \((-\infty, 1)\) is expressed as \(x < 1\) in inequality notation and shown on a number line with an open circle at 1, shading to the left.

Step by step solution

01

Understand the Interval

The given interval is \((-\infty, 1)\). This interval includes all numbers that are less than 1. The round parenthesis means that the number 1 is not included in the interval.
02

Express in Inequality Notation

To express the interval \((-\infty, 1)\) in inequality notation, you state that any number \(x\) within the interval satisfies the condition \(x < 1\).
03

Draw the Number Line

To represent the interval on a number line, draw a horizontal line and locate the point 1 on it. Shade the line to the left of 1, indicating that all numbers less than 1 are included in the interval. Use an open circle at 1 to show that 1 is not included.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Intervals
Intervals are a way of expressing a set of numbers between two points. They are incredibly useful in mathematics, especially when dealing with ranges of numbers.Different types of brackets are used to indicate whether endpoints are included.
  • Round brackets, like \( ( \) or \( ) \), indicate that the endpoint is not included. This is known as an open interval.
  • Square brackets, like \[ [ \text{or} ] \] , mean the endpoint is included, known as a closed interval.
The interval \((-\infty, 1)\) is open because it uses round brackets, excluding 1 from the set.Intervals can be finite or infinite. The infinite aspect here is shown by \(-\infty\), which denotes that the numbers can continue indefinitely in one direction. Hence, the phrase \'from negative infinity to 1\' describes a set of all numbers less than 1.
Number Line
A number line is a simple and vital tool for visualizing numbers. It helps us understand intervals and inequalities quickly.To draw a number line:
  • Start with a horizontal line.
  • Mark specific points, like zero and integers, to orient yourself.
  • Locate and mark the specific value related to your interval. In our problem, this value is 1.
For the given interval \((-\infty, 1)\), you would:
  • Place an open circle at 1 to indicate it is not included.
  • Shade the line to the left of 1, representing all numbers less than 1.
The number line provides a visual representation of our interval, which can make comprehension of abstract concepts clearer.
Mathematical Notation
Mathematical notation is a system of symbols used to write mathematical concepts and relationships in a clear and concise way.In our exploration of the interval \((-\infty, 1)\), we have encountered several important symbols:
  • The round brackets \( ( ) \) specify open intervals where the boundary points are not part of the interval.
  • The infinity symbol \(\infty \) is used to indicate continuation indefinitely in mathematics. Negative infinity goes in the opposite direction.
  • Inequality symbols, such as \( < \), express relationships between numbers. The notation \( x < 1 \) tells us all the numbers of \( x \) are less than 1.
Using this concise notation method, complex ideas become more manageable and communicable for mathematicians globally.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Use a graphing utility to graph the following three parallel lines in the standard viewing rectangle: \(y+4=-0.5(x-2) ; y-3=-0.5(x+2) ; y=-0.5 x\) (b) Experiment with different settings for \(\mathrm{Xmin}, \mathrm{Xmax}\) Ymin, and Ymax. In each case, do the three lines still appear to be parallel?

Verify the identity $$\left(y_{2}-y_{1}\right) /\left(x_{2}-x_{1}\right)=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right)$$ What does this identity tell you about calculating slope?

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(x\) and 1 is at least \(1 / 2\).

Find the standard equation of the circle tangent to the \(x\) -axis and with center \((3,5) .\) Hint: First draw a sketch.

Find an equation for the line that is described, and sketch the graph. Write the answer in the form \(A x+B y+C=0\). Passes through (-3,4) and is parallel to the \(x\) -axis.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.