/*! 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 45 Solve the equations and inequali... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 45

Solve the equations and inequalities. $$\frac{x+1}{2}+x=6$$

Short Answer

Expert verified
x = \( \frac{11}{3} \)

Step by step solution

01

Eliminate the Fraction

First, eliminate the fraction by multiplying every term by 2 to clear the denominator: \[ 2 \times \frac{x+1}{2} + 2 \times x = 2 \times 6 \] This simplifies to: \[ x + 1 + 2x = 12 \]
02

Combine Like Terms

Next, combine the like terms on the left side of the equation: \[ x + 2x + 1 = 12 \] This simplifies to: \[ 3x + 1 = 12 \]
03

Isolate the Variable

Subtract 1 from both sides to isolate the term with the variable: \[ 3x + 1 - 1 = 12 - 1 \] This simplifies to: \[ 3x = 11 \]
04

Solve for x

Finally, divide both sides by 3 to solve for \( x \): \[ x = \frac{11}{3} \]

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.

Eliminating Fractions
When solving equations with fractions, it's often easier to eliminate the fractions first. This step helps simplify the equation and makes it more straightforward to solve. To eliminate a fraction, you can multiply every term in the equation by the denominator of the fraction. For example, in the equation \( \frac{x+1}{2}+x=6 \), the denominator is 2.
We multiply every term by 2 to clear the fraction:
\[ 2 \times \frac{x+1}{2} + 2 \times x = 2 \times 6 \] This simplifies to: \[ x + 1 + 2x = 12 \] By getting rid of the fraction, the equation is now easier to manage!
Combining Like Terms
After simplifying the equation by eliminating fractions, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In the simplified equation \( x + 1 + 2x = 12 \), the like terms are \( x \) and \( 2x \).
We combine these by adding their coefficients:
\[ x + 2x + 1 = 12 \] This simplifies to:
\[ 3x + 1 = 12 \] Combining like terms helps make the equation more condensed and easier to solve.
Isolating Variables
At this stage, we need to isolate the variable on one side of the equation. This usually involves moving all the terms that do not contain the variable to the other side of the equation. In the example \( 3x + 1 = 12 \), we want to isolate \( 3x \).
To do this, we subtract 1 from both sides:
\[ 3x + 1 - 1 = 12 - 1 \] Which simplifies to:
\[ 3x = 11 \] Isolating the variable helps you focus on solving for that variable.
Solving for x
The final step is to solve for the variable. Once we have the variable isolated, we can easily solve for it by performing the necessary arithmetic operations. From the previous example, we have \( 3x = 11 \).
To solve for \( x \), we divide both sides by 3:
\[ x = \frac{11}{3} \] This gives us the solution:
\[ x = \frac{11}{3} \] Solving for \( x \) gives us the final answer to the equation.

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

Solve the equations and inequalities. $$5(y-3)+\frac{2-y}{3}=\frac{7 y+1}{4}$$

Solve the equations and inequalities. $$\frac{t}{2}+\frac{t-1}{3}+\frac{t-6}{4}=t-2$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{3 x+2}{15}+\frac{x}{40}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. How much water should be added to a radiator that contains 10 gallons of an \(80 \%\) antifreeze solution to dilute it to a \(50 \%\) antifreeze solution? [Hint: Pure water is \(0 \% \text { antifreeze. }]\)

Solve the equations and inequalities. $$\frac{w-2}{6}=\frac{w-1}{9}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics 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.