/*! 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 7 Solve. $$\frac{5}{3 x+2}=\frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Solve. $$\frac{5}{3 x+2}=\frac{3}{2 x}$$

Short Answer

Expert verified
The solution is \( x = 6 \).

Step by step solution

01

Cross-multiply

To eliminate the fractions, cross-multiply the two sides of the equation. This gives you:\[ 5 \times 2x = 3 \times (3x + 2) \]This simplifies to:\[ 10x = 9x + 6 \]
02

Isolate the variable

Subtract \(9x\) from both sides of the equation to isolate the variable on one side:\[ 10x - 9x = 6 \]This simplifies to:\[ x = 6 \]
03

Verify the solution

Substitute \(x = 6\) back into the original equation to check if both sides are equal:\[ \frac{5}{3(6) + 2} = \frac{3}{2(6)} \]This simplifies to:\[ \frac{5}{18 + 2} = \frac{3}{12} \]\[ \frac{5}{20} = \frac{3}{12} \]\[ \frac{1}{4} = \frac{1}{4} \]So, the solution \(x = 6\) is verified.

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.

cross-multiplication
To solve rational equations, one of the most useful techniques is cross-multiplication. This helps to eliminate fractions and simplify the equation into a more manageable form.
For example, in the given problem, we start with:
\(\frac{5}{3x+2}=\frac{3}{2x} \)
Using cross-multiplication, we multiply each numerator by the denominator of the opposite fraction:
\[ 5 \times 2x = 3 \times (3x + 2) \] This results in a new equation without fractions:
\[ 10x = 9x + 6 \]
Cross-multiplication is a powerful tool because it converts a rational equation into a polynomial equation, making it easier to solve.
isolating the variable
After cross-multiplying, the next crucial step is isolating the variable. This step involves getting the variable on one side of the equation to solve for it directly.
In our specific example, after cross-multiplying, we have:
\[ 10x = 9x + 6 \]
To isolate \(x\), we subtract \(9x\) from both sides:
\[ 10x - 9x = 6 \]
Simplifying this, we get:
\[ x = 6 \]
Isolating the variable is crucial because it lets us find the value of \(x\) that satisfies the equation.
verifying solutions
After solving for the variable, it's important to verify the solution. This ensures that the solution we found is correct and satisfies the original equation.
For the given problem, we found \( x = 6 \). To verify, we substitute this back into the original equation:
\[ \frac{5}{3(6) + 2} = \frac{3}{2(6)} \]
Simplifying both sides:
\[ \frac{5}{18 + 2} = \frac{3}{12} \]
\[ \frac{5}{20} = \frac{3}{12} \]
\[ \frac{1}{4} = \frac{1}{4} \]
Both sides are equal, which confirms that our solution \( x = 6 \) is correct. Verifying solutions is essential to confirm the accuracy of your answer and ensure there are no mistakes.

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. $$|x+2| \leq|x-5|$$

Solve and write interval notation for the solution set. Then graph the solution set. $$|7-x| \geq-4$$

The cost of a 30 -sec Super Bowl ad has increased more than \(70 \%\) since \(2004 .\) The function $$C(x)=0.17 x+2.25$$ can be used to estimate the cost of a 30 -sec ad, in millions of dollars, \(x\) years after 2004 (Source: Katar Media ). Use this function for Exercises 127 and \(128\). Estimate the cost of a 30 -sec Super Bowl ad in 2014 .

Solve. \(\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}},\) for \(T_{1}\) (A chemistry formula for gases)

Solve and write interval notation for the solution set. Then graph the solution set. $$\left|\frac{2 x-1}{3}\right| \geq \frac{5}{6}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.