/*! 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 24 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 24

Simplify each expression. $$ \frac{7}{k-2}-\frac{5}{2(k-2)} $$

Short Answer

Expert verified
\(\frac{9}{2(k-2)}\)

Step by step solution

01

Identify a common denominator

Observe that the denominators are different. The first fraction has the denominator \(k-2\), and the second fraction has the denominator \(2(k-2)\). The common denominator is \(2(k-2)\).
02

Adjust the first fraction

To combine the fractions, we need to rewrite the first fraction with the common denominator. Multiply the numerator and the denominator of the first fraction by 2: \(\frac{7}{k-2} \times \frac{2}{2} = \frac{14}{2(k-2)}\).
03

Subtract the fractions

Now that both fractions have the same denominator, subtract the second fraction from the first: \(\frac{14}{2(k-2)} - \frac{5}{2(k-2)}\). Subtract the numerators and keep the common denominator: \(\frac{14-5}{2(k-2)} = \frac{9}{2(k-2)}\).
04

Simplify the expression

The simplified expression is \(\frac{9}{2(k-2)}\).

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.

Common Denominator
When dealing with algebraic fractions, identifying a common denominator is essential. It allows us to combine fractions that have different denominators. In the given exercise, we have two fractions with denominators of \(k-2\) and \(2(k-2)\). To achieve a common denominator, we should look for the least common multiple of these denominators. Thankfully, since \(2(k-2)\) already includes \(k-2\), our common denominator is simply \(2(k-2)\). Now, both fractions can be expressed using this common denominator, making further operations straightforward.
Fraction Subtraction
Subtraction of fractions is slightly trickier than addition. Once the fractions share a common denominator, you only subtract the numerators while keeping the denominator consistent. In the example, after converting the first fraction to have the same denominator as the second, we proceed to subtract the numerators: \(14 - 5\). This results in \(9\), which we then place over our common denominator of \(2(k-2)\). Hence, we get the combined fraction \(\frac{9}{2(k-2)}\).
Algebraic Fractions
Algebraic fractions resemble numerical fractions, but they contain variables in their numerators, denominators, or both. Handling algebraic fractions requires the same principles as numerical fractions. However, special attention is necessary when variables are involved as denominators cannot equal zero. For example, in \(\frac{7}{k-2}\), \(k\) cannot be \(2\) because it would make the denominator zero, causing the fraction to be undefined. Always consider these restrictions to avoid mathematical errors.
Simplification Steps
At the core of simplifying algebraic expressions are concise, clear steps:
  • Identify a common denominator for all fractions involved.
  • Adjust the numerators to reflect this common denominator.
  • Perform the subtraction or addition operation on the numerators.
  • Combine the result into a single fraction with the common denominator.
  • Simplify the final expression, if possible.
Taking the example given:
  • We identified \(2(k-2)\) as the common denominator.
  • Adjusted the first fraction by multiplying by the necessary factor to match this denominator.
  • Subtracted the numerators \(14 - 5\) to get \(9\).
  • Placed this result over the common denominator to get \(\frac{9}{2(k-2)}\).
This systematic approach ensures accuracy and clarity in solving algebraic fractions.

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

Factor each expression. $$ -3 x^{2}+3 x+18 $$

Solve each inequality, and graph the solution on the number line. $$ 3 x-9<-4.5 x+6 $$

Factor each expression. $$ 0.5 a^{2}-2 a-16 $$

Sports A volleyball league has two divisions. At the end of the season, Team \(\mathrm{A}\) is in first place in Division 1 , followed by Team \(\mathrm{B}\) and Team \(\mathrm{C}\) . Team \(\mathrm{D}\) is in first place in Division 2 , followed by Team \(\mathrm{E}\) and Team \(\mathrm{F}\) . League organizers have structured the playoffs as shown below. a. Model this series six times, keeping a tally of the champion teams. Flip a coin or roll a die to determine the winner of each game. b. Which team won the most championships? c. Determine the probability that each team will win the champ-win. (Hint: Assume both teams in each game are equally likely to win. (Hint: Imagine that a series is played eight times, and determine how many times you can expect each team to win the championship.)

Kai wants to create a fair dice game for two people, using divisibility by \(3 .\) He decided on these rules: \(\cdot\) Each player rolls one die. \(\cdot\) The players find the sum of the two numbers. \(\cdot\) Player 1 scores if the sum is divisible by \(3 .\) Player 2 scores if the sum is not divisible by \(3 .\) Kai isn't sure how many points each player should score each time. a. Make a table showing the probability of each sum being rolled. b. What is the probability that Player 1 scores on a given roll? c. Kai decided to give 2 points to Player 2 when the sum isn't divisible by \(3 .\) For the game to be fair, how many points should Player 1 score each time the sum is divisible by 3\(?\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.