/*! 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 72 Simplify. $$ \frac{3}{x-2}-\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 72

Simplify. $$ \frac{3}{x-2}-\frac{x+1}{x} $$

Short Answer

Expert verified
\[ \frac{-x^2 + 4x + 2}{x(x-2)} \]

Step by step solution

01

Find a Common Denominator

To subtract these fractions, they must have a common denominator. The denominators are \(x-2\) and \(x\). The common denominator will be \(x(x-2)\).
02

Adjust First Fraction

The first fraction is \(\frac{3}{x-2}\). To get a common denominator \(x(x-2)\), multiply the numerator and the denominator by \(x\). This gives: \frac{3 \cdot x}{(x-2) \cdot x} = \frac{3x}{x(x-2)}\.
03

Adjust Second Fraction

The second fraction is \frac{x+1}{x}\. To get a common denominator \(x(x-2)\), multiply the numerator and the denominator by \((x-2)\). This gives: \frac{(x+1)(x-2)}{x(x-2)}\.
04

Combine the Fractions

Now, we have: \[\frac{3x}{x(x-2)} - \frac{(x+1)(x-2)}{x(x-2)}\] Since the denominators are the same, subtract the numerators: \[\frac{3x - (x+1)(x-2)}{x(x-2)}\]
05

Expand and Simplify

Expand \( (x+1)(x-2) \): \[(x+1)(x-2) = x^2 - 2x + x - 2 = x^2 - x - 2\]Now, substitute this into the expression: \[\frac{3x - (x^2 - x - 2)}{x(x-2)}\] Combine like terms: \[3x - x^2 + x + 2 = -x^2 + 4x + 2\]Giving us the new numerator: \[\frac{-x^2 + 4x + 2}{x(x-2)}\]

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.

Fraction Subtraction
Subtracting fractions can seem tricky, but it becomes easier with a step-by-step approach.
Fractions must have the same denominator before you can subtract them.
Once they share a common denominator, you can combine them easily. In our exercise, the fractions are \(\frac{3}{x-2}\) and \(\frac{x+1}{x}\).
At first, these fractions do not have a common denominator.
That's why we need the next core concept: common denominators.
Common Denominators
A common denominator is essential to combine fractions.
The common denominator of two fractions is a multiple of their denominators. Here, the denominators are \((x-2)\) and \((x)\). The least common multiple (LCM) of these is \((x(x-2))\).
By adjusting each fraction to have this new common denominator, we can simplify the calculation:
1. For \(\frac{3}{x-2}\), multiply the numerator and denominator by \(x\), getting \(\frac{3x}{x(x-2)}\).
2. For \(\frac{x+1}{x}\), multiply the numerator and denominator by \((x-2)\), getting \(\frac{(x+1)(x-2)}{x(x-2)}\).
Now both fractions are ready to be subtracted.
Expanding Expressions
Expressions often need expanding to simplify them further.
In our case, after adjusting the fractions to a common denominator, we need to subtract the numerators.
That means we need to handle \((x+1)(x-2))\):
- First, expand \((x+1)(x-2)\) to get \(x^2 - x - 2\).
- Next, combine like terms in the final step: Subtract the expanded expression from the first numerator: 3x - (x^2 - x - 2).
Simplifying this, we get \(-x^2 + 4x + 2\).
The last step is to place this new numerator over the original common denominator: \(\frac{-x^2 + 4x + 2}{x(x-2)}\).
Here's the fully simplified expression!

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 \(V(t)=750(1.2)^{t}\) is used to predict the value \(V(t)\) of a certain rare stamp \(t\) years after \(2016 .\) Do not calculate \(V^{-1}(t),\) but explain how \(V^{-1}\) could be used.

Simplify. \(\left(-3 x^{-6} y^{4}\right)^{-2}\)

Solve for \(x\). Give an approximation to four decimal places. $$ \log (492 x)=5.728 $$

Solve for \(x\). Give an approximation to four decimal places. $$ \log \left(275 x^{2}\right)=38 $$

Solve $$ \log _{2} x=-1 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.