/*! 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 52 Solve each equation. $$\frac{x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 52

Solve each equation. $$\frac{x-2}{2}-\frac{x-3}{4}=\frac{7}{4}$$

Short Answer

Expert verified
x = 8

Step by step solution

01

Combine Like Terms

First, we'll find a common denominator for the fractions. The common denominator for 2 and 4 is 4. Thus, rewrite the fraction \(\frac{x-2}{2}\) as \(\frac{2(x-2)}{4}\). The equation becomes: \[\frac{2(x-2)}{4} - \frac{x-3}{4} = \frac{7}{4}\]
02

Simplify the Equation

Combine the fractions on the left-hand side: \[\frac{2(x-2) - (x-3)}{4} = \frac{7}{4}\]
03

Combine Numerators

Distribute and combine the numerators: \[2(x-2) - (x-3) = 2x - 4 - x + 3 = x - 1\]. Now, the equation is: \[\frac{x - 1}{4} = \frac{7}{4}\]
04

Cross Multiply

Since the denominators are the same, set the numerators equal to each other: \[x - 1 = 7\]
05

Solve for x

Add 1 to both sides to isolate x: \[x = 8\]

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 Denominators
To solve equations involving fractions, finding a common denominator is key. This means identifying a number that all denominators can divide into evenly. In our example, the fractions have denominators of 2 and 4. The smallest common denominator for these is 4. Here's how to do it:

  • Convert \(\frac{x-2}{2}\) into \(\frac{2(x-2)}{4}\). Notice we multiply both the numerator and the denominator by 2 to ensure the denominator is 4.
  • Now, all fractions in the equation share a common denominator: \(\frac{2(x-2)}{4} - \frac{x-3}{4} = \frac{7}{4}\).
Having a common denominator simplifies combining fractions and solving equations.
Combining Like Terms
Once fractions have a common denominator, combining like terms is simpler. Like terms are terms that have the same variable raised to the same power. Here's a step-by-step process:

  • Start by combining the fractions: \(\frac{2(x-2)}{4} - \frac{x-3}{4}\).
  • Express both terms over the same denominator: \(\frac{2(x-2) - (x-3)}{4}\).
  • Now distribute and combine: 2(x-2) becomes 2x - 4 and - (x-3) becomes -x + 3.
  • Combine these to form the numerator: 2x - 4 - x + 3 = x - 1.
Combining like terms ensures we've simplified our equation to \(\frac{x - 1}{4} = \frac{7}{4}\).
Cross Multiplication
With common denominators, cross multiplication becomes straightforward. If we have \(\frac{a}{b} = \frac{c}{d}\) and both sides have the same denominator, we can simplify to \a = c\. Using cross multiplication:

  • Since \(\frac{x - 1}{4} = \frac{7}{4}\), set the numerators equal: \x - 1 = 7\.
  • This is because the fractions are equal, implying their numerators are also equal.
  • Now solve for x by adding 1 to both sides: \x - 1 + 1 = 7 + 1\.
  • This simplifies to \x = 8\.
Cross multiplication makes solving equations involving fractions easier once the common denominators are established.

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

Discussion. For which of the inequalities is the notation used correctly? a) \(-2 \leq x<3\) b) \(-4 \geq x<7\) c) \(-1 \leq x>0\) d) \(6

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$1-3(x-2)=4(x-1)-3$$

Show a complete solution to each problem. Mixed nuts. Cashews sell for \(\$ 1.20\) per quarter-pound, and Brazil nuts sell for \(\$ 1.50\) per quarter-pound. How many pounds of cashews should be mixed with 20 pounds of Brazil nuts to get a mix that sells for \(\$ 1.30\) per quarterpound?

Show a complete solution to each problem. Alcohol solutions. Amy has two solutions available in the laboratory, one with \(5 \%\) alcohol and the other with \(10 \%\) alcohol. How much of each should she mix together to obtain 5 gallons of an \(8 \%\) solution?

Show a complete solution to each problem. Stacy's square. Stacy has 70 meters of fencing and plans to make a square pen. In one side she is going to leave an opening that is one-half the length of the side. If she uses all 70 meters of fencing, how large can the square be?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.