/*! 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 36 Solve and check. Label any contr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 36

Solve and check. Label any contradictions or identities. $$4 x-x=2 x+x$$

Short Answer

Expert verified
All values of \(x\) are solutions. This is an identity.

Step by step solution

01

Simplify both sides of the equation

Combine like terms on both sides of the equation. On the left side: \(4x - x = 3x\). On the right side: \(2x + x = 3x\).
02

Set the simplified forms equal

After simplifying, the equation becomes: \(3x = 3x\).
03

Analyze for contradictions or identities

Notice that \(3x = 3x\) is an identity because both sides are always equal no matter the value of \(x\).
04

State the solution

Since it's an identity, all values of \(x\) are solutions to this equation.

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.

like terms
In algebra, solving equations often involves simplifying expressions by combining 'like terms'. Like terms are terms that have the same variables raised to the same powers. For example, in the expression \(4x - x\), both \(4x\) and \(x\) are like terms because they share the variable \(x\). By summing these coefficients, we simplify \(4x - x\) to \(3x\).
This operation helps in making the equations simpler and more manageable. To properly identify and combine like terms, look for:
  • Coefficients with identical variables
  • Variables raised to the same power

This is a fundamental skill, as it streamlines expressions and paves the way to easier calculations.
identities in algebra
Identities in algebra refer to equations that are always true, regardless of the variable鈥檚 value. Once we simplified the original equation \(4x - x = 2x + x\), we derived \(3x = 3x\). This equation is a clear example of an identity. It states a truth that applies universally: both sides of the equation are inherently and permanently equal.
Recognizing identities involves:
  • Simplifying both sides of an equation
  • Observing if one side mirrors the other

Since the equation holds for any value of \(x\), there's no single solution; instead, the solution set includes all real numbers.
linear equations
Linear equations are algebraic equations in which the highest power of the variable is one. They form straight lines when graphed. The general form of a linear equation is \(ax + b = 0\), where \(a\) and \(b\) are constants.
To solve linear equations, follow these steps:
  • Combine like terms on both sides
  • Simplify the equation to isolate the variable

For instance, our original exercise was \(4x - x = 2x + x\). Simplifying both sides results in \(3x = 3x\). This is a linear equation, although in this case, it turns out to be an identity, meaning it is true for all \(x\).

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

Barnes \& Noble offers a member card for \(\$ 25\) per year. This card entitles a customer to a \(40 \%\) discount off list price on hardcover bestsellers, a \(20 \%\) discount on adult hardcovers, and a \(10 \%\) discount on other purchases. Describe two sets of circumstances for which an individual would save money by becoming a member.

Review operations with real numbers. Simplify. $$ -2+(-5)-7 $$

The perimeter of a rectangle is \(101.74 \mathrm{cm} .\) If the length is \(4.25 \mathrm{cm}\) longer than the width, find the dimensions of the rectangle.

Use an inequality and the five-step process to solve each problem. Most insurance companies will replace a vehicle if an estimated repair exceeds \(80 \%\) of the "blue-book" value of the vehicle. Michelle's insurance company paid \(\$ 8500\) for repairs to her Subaru after an accident. What can be concluded about the blue-book value of the car?

Wedding costs. The Arnold Inn offers two plans for wedding parties. Under plan A, the inn charges \(\$ 30\) for each person in attendance. Under plan \(\mathrm{B}\), the inn charges \(\$ 1300\) plus \(\$ 20\) for each person in excess of the first 25 who attend. For what size parties will plan \(\mathrm{B}\) cost less? (Assume that more than 25 guests will attend.)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.