Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 3) = 1/4
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 107
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).
Verified step by step guidance1
Let the number be represented by the variable \(x\).
Translate the problem statement into an equation: "5 times a number decreased by 4" becomes \(5x - 4\), and "the principal square root of this difference" is \(\sqrt{5x - 4}\). This is said to be 2 less than the number, so we write \(\sqrt{5x - 4} = x - 2\).
To eliminate the square root, square both sides of the equation: \((\sqrt{5x - 4})^2 = (x - 2)^2\), which simplifies to \(5x - 4 = (x - 2)^2\).
Expand the right side: \((x - 2)^2 = x^2 - 4x + 4\), so the equation becomes \(5x - 4 = x^2 - 4x + 4\).
Rearrange all terms to one side to form a quadratic equation: \(0 = x^2 - 4x + 4 - 5x + 4\), which simplifies to \(0 = x^2 - 9x + 8\). Then, solve this quadratic equation for \(x\) using factoring, completing the square, or the quadratic formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Formulating Algebraic Equations from Word Problems
This involves translating a verbal description into a mathematical equation. Key phrases like '5 times a number' and 'decreased by 4' guide the construction of expressions, enabling the problem to be solved algebraically.
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Introduction to Exponents
Square Roots and Principal Square Root
The principal square root of a number is the non-negative root. Understanding this helps interpret expressions like 'the principal square root of this difference,' ensuring correct handling of square root operations in equations.
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02:20Imaginary Roots with the Square Root Property
Solving Quadratic Equations
After forming the equation, solving it often leads to a quadratic equation. Techniques such as factoring, completing the square, or using the quadratic formula are essential to find the number(s) that satisfy the problem.
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06:08Solving Quadratic Equations by Factoring
Related Practice
Textbook Question
In Exercises 107–110, use graphs to find each set. (-2,1] ∩ [-1,3)
Textbook Question
Solve each equation. - 2{7 - [4 -2(1 - x) + 3]} = 10 - [4x - 2(x - 3)]
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.
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 2x/(x - 3) + 6/(x + 3) = - 28/(x2 - 9)
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Textbook Question
Use the table to solve each inequality. - 3 < 2x - 5 ≤ 3
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