Skip to main content
Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 103
Capitolo 9, Problema 103

Solve: x/(x−3)=2x/(x−3)−5/3

Guida verificata passo dopo passo
1
Start by observing the equation: \(\frac{x}{x-3} = \frac{2x}{x-3} - \frac{5}{3}\). Notice that the denominators on the left and right sides involve \(x-3\), so we should consider the domain restriction \(x \neq 3\) to avoid division by zero.
To eliminate the denominators, multiply both sides of the equation by the least common denominator (LCD). The denominators are \(x-3\) and 3, so the LCD is \(3(x-3)\). Multiply every term by \(3(x-3)\):
\[3(x-3) \cdot \frac{x}{x-3} = 3(x-3) \cdot \frac{2x}{x-3} - 3(x-3) \cdot \frac{5}{3}\]
Simplify each term by canceling the denominators:
\[3x = 3 \cdot 2x - 5(x-3)\]
Now, expand and simplify the right side:
\[3x = 6x - 5x + 15\]
Combine like terms on the right side:
\[3x = (6x - 5x) + 15 = x + 15\]
Finally, isolate \(x\) by subtracting \(x\) from both sides:
\[3x - x = 15\]
Simplify the left side:
\[2x = 15\]
At this point, you can solve for \(x\) by dividing both sides by 2.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, identify common denominators and eliminate fractions by multiplying both sides accordingly. This simplifies the equation to a polynomial form, making it easier to solve.
Video consigliato:
05:56
Introduction to Rational Equations

Restrictions on the Domain

When solving equations with variables in denominators, certain values can make the denominator zero, which is undefined. Identifying these restrictions is crucial to exclude invalid solutions from the final answer.
Video consigliato:
3:51
Domain Restrictions of Composed Functions

Combining Like Terms and Simplifying

After clearing denominators, combine like terms on each side to simplify the equation. This step helps isolate the variable and solve the resulting linear or polynomial equation efficiently.
Video consigliato:
03:50
Adding & Subtracting Like Radicals
Pratica correlata
Domanda del libro di testo

Use the formula an=4+(n−1)(−7)a_n=4+(n-1)(-7) to find the eighth term of the sequence 4,−3,−10,…4, −3, −10,…

821
views
Domanda del libro di testo

The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of a flood for three consecutive years?

634
views
Domanda del libro di testo

Write the first five terms of the sequence whose first term is 9 and whose general term is

for n≥2.

823
views
Domanda del libro di testo

The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of no flooding for four consecutive years?

773
views