/*! 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 55 Solve each equation. See Section... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 55

Solve each equation. See Sections 2.3 and 6.6 $$ 3 x-4=3(x+1) $$

Short Answer

Expert verified
The equation has no solution.

Step by step solution

01

Distribute the right side of the equation

Start by distributing the 3 on the right side of the equation. This means you will multiply 3 by each term inside the parentheses. The equation becomes:\[ 3x - 4 = 3x + 3 \]
02

Subtract 3x from both sides

To isolate the variable, subtract 3x from both sides of the equation. This helps in simplifying the terms involving x. This results in:\[ -4 = 3 \]
03

Analyze the simplified equation

After simplifying the equation, we end up with \(-4 = 3\). This is not a true statement, indicating that there are no values of \(x\) that will satisfy the original 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.

Distributive Property
The distributive property is a fundamental algebraic principle used to simplify expressions. It allows us to multiply a single term outside the parentheses by each term inside the parentheses. For the equation \( 3(x+1) \), the distributive property comes into play.

Let's see how it works:
  • Place the number outside the parentheses in this case, 3, and distribute it to each term inside the parentheses.
  • So, 3 is multiplied by both \( x \) and \( 1 \) separately.
This results in the expression \( 3x + 3 \). The beauty of the distributive property is that it simplifies equations, making it easier to compare expressions on each side of the equation.

In our problem, after applying the distributive property, the equation becomes \( 3x - 4 = 3x + 3 \). This is a crucial first step towards solving the equation.
Isolating the Variable
Isolating the variable refers to the steps taken to get all instances of the variable on one side of the equation. The goal is to make it possible to determine the value of the unknown variable.

In the exercise, after using the distributive property, the next step was to isolate the variable \( x \) by subtracting \( 3x \) from both sides.
  • This operation aims to remove \( x \) from one side and simplify the equation further.
  • On subtracting \( 3x \) from both sides, we surprisingly find that there are no longer any \( x \) terms left on either side of the equation.
This results in the equation \( -4 = 3 \). The purpose of isolating the variable is to solve for the variable, but in this case, it leads us to an interesting conclusion about the equation itself.
No Solution Equations
Sometimes, when simplifying an equation, you might end up with something peculiar like \( -4 = 3 \) which clearly is a false statement. This indicates that the equation is a "no solution equation."
  • When an equation simplifies to a false statement, it implies there are no values of \( x \) that can satisfy the equation.
  • This happens when the expressions on both sides of the initial equation are parallel and never intersect.
Such equations are important as they tell us there's a fundamental inconsistency in the relationship described. In the given exercise, the use of distributive property and attempts to isolate \( x \) helped identify the problem as one with no solution, highlighting an essential aspect of solvable equations.

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

Without using a calculator, explain which of \(\log 50\) or \(\ln 50\) must be larger and why.

Solve each equation for \(x .\) See Example 4. $$125^{x-2}=25$$

Evaluate each of the following. $$ 9^{-3 / 2} $$

Solve each equation for \(x\). See Sections 2.3 and 6.6. $$ 4 x-8 y=10 x $$

Solve each equation. See Sections 2.3 and 6.6 $$ 3 x-7=11 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.