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 105a
Solve each equation. 4x + 13 - {2x - [4(x - 3) - 5]} = 2(x - 6)
Verified step by step guidance1
Distribute the terms inside the brackets and parentheses. Start with the innermost parentheses: expand \(4(x - 3)\) to \(4x - 12\). Then simplify \([4x - 12 - 5]\) to \(4x - 17\). Finally, simplify \{2x - (4x - 17)\} to \{2x - 4x + 17\}, which becomes \(-2x + 17\).
Substitute the simplified expression \(-2x + 17\) back into the original equation. The equation becomes \(4x + 13 - (-2x + 17) = 2(x - 6)\).
Simplify the left-hand side of the equation by combining like terms: \(4x + 13 + 2x - 17\) simplifies to \(6x - 4\). The equation now reads \(6x - 4 = 2(x - 6)\).
Distribute the \(2\) on the right-hand side: \(2(x - 6)\) becomes \(2x - 12\). The equation now reads \(6x - 4 = 2x - 12\).
Isolate \(x\) by first subtracting \(2x\) from both sides: \(6x - 2x - 4 = -12\), which simplifies to \(4x - 4 = -12\). Then add \(4\) to both sides: \(4x = -8\). Finally, divide both sides by \(4\) to solve for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distributive Property
The distributive property states that a(b + c) = ab + ac. This property is essential for simplifying expressions where a term is multiplied by a sum or difference. In the given equation, applying the distributive property will help in expanding terms like 2(x - 6) and 4(x - 3) effectively.
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Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This step is crucial in simplifying equations to make them easier to solve. In the equation provided, after applying the distributive property, you will need to combine terms involving 'x' and constant terms to isolate the variable.
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Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. This typically includes isolating the variable on one side of the equation through operations such as addition, subtraction, multiplication, or division. Understanding the steps to isolate 'x' in the given equation is key to finding the solution.
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Related Practice
Textbook Question
Solve each equation. - 2{7 - [4 -2(1 - x) + 3]} = 10 - [4x - 2(x - 3)]
Textbook Question
Use the table to solve each inequality. - 3 < 2x - 5 ≤ 3
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Textbook Question
In Exercises 104–106, express each interval in set-builder notation and graph the interval on a number line. (-2, ∞)
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 2) = 1/3
Textbook Question
In Exercises 101–106, solve each equation.