/*! 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 80 Based on material learned earlie... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 80

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(\frac{1}{3}(x-6)+4 x>0\)

Short Answer

Expert verified
\( x > \frac{3}{13} \text{,or approximately x} > 0.462.

Step by step solution

01

Distribute the fraction

Distribute \(\frac{1}{3}\) through \((x - 6)\). This gives \(\frac{1}{3}x - 2\).
02

Combine like terms

Combine \(\frac{1}{3}x\) and \(+4x\). This simplifies to \( \frac{1}{3}x + 4x - 2 > 0 \).
03

Simplify the equation

Convert \(\frac{1}{3}x\) to decimals by finding a common denominator. \( \frac{1}{3}x\text{ is the same as }x/3 \). So, it can be rewritten as: \( \frac{1}{3}x = 0.3333x \text{ or } 0.3333x \). Combine with \(4x\) to get: \( \0.3333x + 4x \).
04

Solve for x

Combine terms and solve for x: \(4.3333x > 2 \). Divide both sides by 4.3333: \(\frac{2}{4.3333}= x \). This simplifies to \(\frac{2}{4.33} \text{is approximately equal to } x \)

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.

solving linear inequalities
Solving linear inequalities is a fundamental aspect of algebra. It involves finding the values of a variable that satisfy the given inequality. Here, we are tasked with solving \(\frac{1}{3}(x-6)+4x>0\).

To solve this, follow these steps:
  • First, we distribute any fractions or coefficients through parentheses.

  • Next, combine all like terms to simplify the expression.

  • Finally, solve for the variable by isolating it on one side of the inequality.

Notice that solving an inequality is quite similar to solving an equation, but we must remember the crucial rule: if we multiply or divide both sides by a negative number, we must reverse the inequality sign.
combining like terms
Combining like terms allows us to simplify expressions more effectively. Like terms are terms that contain the same variable raised to the same power.

In the inequality \(\frac{1}{3}(x-6)+4x>0\), we distribute \(\frac{1}{3}\) through each term to get \(\frac{1}{3}x - 2\).

Now we have: \(\frac{1}{3}x - 2 + 4x > 0\).
Combine the like terms \(\frac{1}{3}x\) and \(4x\).
Converting \(\frac{1}{3}x\) to decimal form, we get 0.3333x, so we add 0.3333x and 4x:

\(\frac{1}{3}x + 4x \rightarrow 0.3333x + 4x = 4.3333x\). Now, our inequality looks like this: 4.3333x - 2 > 0. By combining like terms, we effectively reduce the complexity of the expression, making it easier to solve.
simplifying expressions
Simplifying expressions involves making them as concise as possible without changing their value. This often involves combining like terms and performing arithmetic operations.

Let's consider the expression we derived: 4.3333x - 2 > 0.

Now, we need to isolate x.
  • Add 2 to both sides to get 4.3333x > 2.

  • Then, divide both sides by 4.3333 to isolate x.

This gives us:
\ x > \frac{2}{4.3333} \
Approximating, we get \ x > \frac{2}{4.33} = 0.461\

Hence, the solution to the inequality is \( x > 0.461 \). Simplifying expressions is crucial for making calculations more manageable, especially when dealing with inequalities.

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

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-15 e^{-0.9 t / 30} \cos \left(\sqrt{\left(\frac{\pi}{3}\right)^{2}-\frac{0.81}{900}} t\right) $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: } \frac{x^{2} \cdot \frac{1}{x}-\ln x \cdot 2 x}{\left(x^{2}\right)^{2}}=0 $$

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function \(d(t)=a \sin (\omega t)\) If a tuning fork for the note A above middle \(\mathrm{C}\) on an even-tempered scale \(\left(A_{4},\right.\) the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find \(\omega\). If the maximum displacement of the end of the tuning fork is 0.01 millimeter, find a function that describes the movement of the tuning fork.

Use the Law of Cosines to prove the identity $$ \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c} $$

Establish the identity: \(\csc \theta-\sin \theta=\cos \theta \cot \theta\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.