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 90a
The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x2 + 3x - 10) - 1/(x2 + x - 6) = 3/(x2 - x - 12)
Verified step by step guidance1
Factorize the denominators of each term in the equation. For example, factorize \(x^2 + 3x - 10\), \(x^2 + x - 6\), and \(x^2 - x - 12\) into their respective factored forms.
Identify the least common denominator (LCD) of the equation by combining all unique factors from the denominators.
Multiply through the entire equation by the LCD to eliminate the denominators. This will leave you with a polynomial equation.
Simplify the resulting polynomial equation by combining like terms and rearranging it into standard form, \(ax^2 + bx + c = 0\).
Solve the simplified polynomial equation using appropriate methods such as factoring, completing the square, or the quadratic formula. Check the solutions in the original equation to ensure they do not make any denominator zero, and classify the equation as an identity, conditional, or inconsistent based on the results.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Types of Equations
In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications helps in determining the nature of the solution set for a given equation.
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Factoring Polynomials
Factoring polynomials is a crucial skill in solving rational equations. It involves expressing a polynomial as a product of its factors, which can simplify the equation and make it easier to solve. For example, the expressions in the denominators of the given equation can be factored to identify common terms and simplify the overall equation.
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Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. When solving equations involving rational expressions, it is essential to find a common denominator to combine the fractions effectively. Additionally, one must be cautious of restrictions on the variable that can arise from the denominators, as these can lead to undefined values.
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Related Practice
Textbook Question
In Exercises 59–94, solve each absolute value inequality.
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Textbook Question
In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] y = 2(x + 2)^2 + 5(x + 2) - 3
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Textbook Question
The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x2 + 36)/(x2 - 9)
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Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. (2x + 3)(x + 4) = 1
Textbook Question
In Exercises 59–94, solve each absolute value inequality. 1 < |2 - 3x|
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