Textbook Question
In Exercises 91–100, find all values of x satisfying the given conditions. y = |2 - 3x| and y = 13
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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 93
Solve each equation in Exercises 83–108 by the method of your choice. (3x - 4)2 = 16
Verified step by step guidance1
Start by recognizing that the equation \((3x - 4)^2 = 16\) is a quadratic equation in the form of a perfect square. To solve, take the square root of both sides of the equation. Remember to include both the positive and negative roots when taking the square root.
After taking the square root, the equation becomes \(3x - 4 = \pm 4\). This means you now have two separate linear equations to solve: \(3x - 4 = 4\) and \(3x - 4 = -4\).
Solve the first equation \(3x - 4 = 4\) by isolating \(x\). Add 4 to both sides to get \(3x = 8\), then divide both sides by 3 to find \(x\).
Solve the second equation \(3x - 4 = -4\) by isolating \(x\). Add 4 to both sides to get \(3x = 0\), then divide both sides by 3 to find \(x\).
Combine the solutions from both equations to express the final solution set. These are the values of \(x\) that satisfy the original equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the properties of quadratic equations is essential for solving them effectively.
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Square Roots
Square roots are the values that, when multiplied by themselves, yield the original number. In the context of the equation (3x - 4)^2 = 16, taking the square root of both sides is a crucial step to isolate the variable. It is important to consider both the positive and negative roots when solving such equations.
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Isolating Variables
Isolating variables involves rearranging an equation to solve for a specific variable. This process often includes using inverse operations, such as addition, subtraction, multiplication, or division, to get the variable alone on one side of the equation. Mastery of this concept is vital for effectively solving equations and understanding their solutions.
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Related Practice
Textbook Question
Solve each equation in Exercises 92–93 by making an appropriate substitution. x^4 - 5x^2 + 4 = 0
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. (2x - 5)(x + 1) = 2
Textbook Question
Solve each absolute value inequality. 4 + |3 - x/3| ≥ 9
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Textbook Question
Solve the equations containing absolute value in Exercises 94–95. |2x+1| = 7
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Textbook Question
In Exercises 91–100, find all values of x satisfying the given conditions.
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