/*! 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 31 Determine the values represented... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 31

Determine the values represented by \(\underline{a}\). $$ |a|=1 $$

Short Answer

Expert verified
The values of \(a\) are 1 and -1.

Step by step solution

01

Understand the Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. In mathematical form, if \(|a| = 1\), it means the distance of \(a\) from zero is 1.
02

Set Up Equations

Based on the definition of absolute value, the equation \(|a| = 1\) translates to two possible equations: 1. \(a = 1\) 2. \(a = -1\)
03

Solve the Equations

Solving the two equations from Step 2 separately:1. If \(a = 1\), then the possible value of \(a\) is 1.2. If \(a = -1\), then the possible value of \(a\) is -1.
04

Conclude the Possible Values

Both equations are valid and therefore both possible values are acceptable solutions. Thus, the values represented by \(a\) are \(a = 1\) and \(a = -1\).

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.

Number Line
A number line is a visual representation that helps us understand numbers and their positions in relation to zero and each other. Imagine a straight horizontal line:
  • Zero is in the center.
  • Positive numbers are placed to the right of zero.
  • Negative numbers are placed to the left of zero.
Each point on this line represents a different number, and the distance between points tells us how far apart these numbers are.
For example, the number 1 is one unit to the right of zero, while -1 is one unit to the left. The number line is perfect for understanding absolute value, as it shows both direction and distance from zero.
Distance from Zero
Distance from zero is a central concept in understanding absolute values. The absolute value of a number is its distance from zero on the number line, without considering direction.
For instance, both 1 and -1 are one unit away from zero. Therefore, they both have an absolute value of 1. This concept makes absolute values useful for measuring the size or magnitude of a number, regardless of whether it is positive or negative. So, when we say \(|a| = 1\), it means 'a' could be 1 step away from zero in either direction.
Solving Equations
Solving equations involving absolute values requires understanding how absolute values work. When given \(|a| = 1\), we need to consider both directions on the number line.
This leads to setting up two separate equations based on the definitions of absolute value:
  • The first equation is \(a = 1\), meaning 'a' is 1 step to the right of zero.
  • The second equation is \(a = -1\), meaning 'a' is 1 step to the left of zero.
Solving these equations individually allows us to conclude that there are two possible solutions, since the absolute value does not favor a direction.
Possible Values
Once the equations \(a=1\) and \(a=-1\) are solved, we identify the possible values for the variable 'a'. Conveniently, both solutions are correct and account for all possibilities on the number line where \(|a|=1\).
In absolute value problems like these, it is important to check both potential solutions. The "solution set" consists of:
  • 'a' can be 1, located one step to the right of zero.
  • Or 'a' can be -1, situated one step to the left of zero.
Thus, the problem concludes with both positive and negative possibilities being valid solutions.

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

Simplify. $$ (3 x-1)^{3} $$

Jerry scored \(90,85,92\), and 76 on the first four algebra exams. What must he score on the fifth exam so that his average is at least 80 ?

Given that the height of a right circular cylinder is equal to the radius of the base, derive a formula for the surface area in terms of the radius of the base.

Simplify. $$ (3 x-2)\left(x^{2}-5 x+2\right) $$

How can we check to see if we have simplified an expression correctly?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.