/*! 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 128 Translate to an algebraic expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 128

Translate to an algebraic expression or equation. A real number is its own reciprocal.

Short Answer

Expert verified
\(x = 1\) or \(x = -1\)

Step by step solution

01

Define the variable

Let the real number be represented by the variable \( x \).
02

Set up the equation

The problem states that a real number is its own reciprocal. The reciprocal of a number \( x \) is \( \frac{1}{x} \). Thus, we set up the equation: \[ x = \frac{1}{x} \]
03

Simplify the equation

To solve for \( x \), multiply both sides of the equation by \( x \) to eliminate the fraction: \[ x \cdot x = 1 \]
04

Solve for the variable

Simplify the left side of the equation: \[ x^2 = 1 \]. To solve for \( x \), take the square root of both sides: \[ x = \pm 1 \]

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.

Reciprocal
A reciprocal of a number is essentially its 'flipped' version. If you have a number, say \( y \), its reciprocal is \( \frac{1}{y} \). This concept is crucial in algebra because it helps to simplify and solve a lot of equations. For examples:
  • The reciprocal of 2 is \( \frac{1}{2} \)
  • The reciprocal of 5 is \( \frac{1}{5} \)
  • The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \)
This is vital to understand especially when an equation asks you to find a value that is equal to its own reciprocal, such as in this exercise.
Equation Solving
Solving an equation involves isolating the variable you are trying to solve for. In this case, the equation given is:
\[ x = \frac{1}{x} \] The objective is to find the value(s) of \( x \) that satisfy this equality. Start by multiplying both sides by \( x \) to get rid of the fraction:
\[ x \times x = 1 \] Simplifying it results in:
\[ x^2 = 1 \] Once here, solving for \( x \) typically involves taking the square root on both sides of the equation. The idea is to simplify step by step until the variable is isolated.
Variable Definition
In algebra, defining variables is a crucial first step. A variable is a symbol, usually a letter, that represents a number we don鈥檛 know yet.
In this exercise, we defined the real number as \( x \). By doing so, you can set up an equation that's easier to work with:
For example, 'A real number is its own reciprocal' can be translated to
\[ x = \frac{1}{x} \] Clearly defining the variable from the start helps to avoid confusion and makes solving the equation straightforward.
Square Root
To solve \( x^2 = 1 \), you need to understand the concept of the square root. The square root of a number \( y \) is a number that, when multiplied by itself, gives \( y \).
For example:
  • The square root of 9 is 3, because \( 3 \times 3 = 9 \)
  • The square root of 16 is 4, because \( 4 \times 4 = 16 \)
Taking the square root of both sides in the equation \( x^2 = 1 \) gives you:
\[ x = \pm 1 \] It's important to include both the positive and negative roots because both are valid solutions. Therefore, both 1 and -1 are solutions to the problem.

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. $$ 8 n^{2}-3 n+2\left(n-4 n^{2}\right) $$

Simplify using a calculator. Round your answer to the nearest thousandth. $$ |13.5+8(-4.7)|^{3} $$

Simplify. $$ 9 t^{4}+7 t-5\left(9 t^{3}-2 t\right) $$

If \(n>0, m>0,\) and \(n \neq m,\) classify each of the following as either true or false. $$ n(-n-m)=-n^{2}+n m $$

Write an equivalent expression without using grouping symbols. $$ -(9 x+1) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.