Textbook Question
If 5 times a number is decreased by 4, the principal square root of this difference is 2 less than the number. Find the number(s).
838
views
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 107a
Solve each equation in Exercises 83–108 by the method of your choice. 2x/(x - 3) + 6/(x + 3) = - 28/(x2 - 9)
Verified step by step guidance1
Rewrite the equation and recognize that the denominator \(x^2 - 9\) is a difference of squares, which can be factored as \((x - 3)(x + 3)\). This will help simplify the equation.
Multiply through by the least common denominator (LCD), which is \((x - 3)(x + 3)\), to eliminate the fractions. Be careful to distribute the LCD to each term in the equation.
Simplify each term after multiplying by the LCD. For example, \(\frac{2x}{x - 3} \cdot (x - 3)(x + 3)\) simplifies to \(2x(x + 3)\), and so on for the other terms.
Combine like terms and simplify the resulting equation. This will likely result in a quadratic equation.
Solve the quadratic equation using factoring, the quadratic formula, or completing the square. Be sure to check for any restrictions on the variable (e.g., \(x \neq 3\) and \(x \neq -3\)) to ensure the solution is valid.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including finding common denominators and simplifying, is crucial for solving equations involving them. In this problem, the presence of rational expressions necessitates careful handling to combine and solve the equation effectively.
Recommended video:
Guided course
02:58
Rationalizing Denominators
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler components (factors) that, when multiplied together, yield the original polynomial. In this equation, recognizing that the denominator x^2 - 9 can be factored into (x - 3)(x + 3) is essential for simplifying the equation and eliminating the fractions.
Recommended video:
Guided course
07:30Introduction to Factoring Polynomials
Finding Common Denominators
Finding a common denominator is a key step in adding or subtracting rational expressions. It allows for the combination of fractions into a single expression. In this equation, identifying the least common denominator (LCD) of the fractions involved will facilitate the elimination of the denominators, making it easier to solve for x.
Recommended video:
Guided course
02:58Rationalizing Denominators
Related Practice
Textbook Question
If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).
728
views
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x2 - 20)/(x2 - 7x + 12)
954
views
Textbook Question
In Exercises 107–110, use graphs to find each set. (-2,1] ∩ [-1,3)
521
views
Textbook Question
When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
1211
views
Textbook Question
When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
858
views