Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
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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 85a
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. 2/x + 1/2 = 3/4
Verified step by step guidance1
Rewrite the equation to eliminate the fractions by finding the least common denominator (LCD) of all the terms. The denominators are x, 2, and 4, so the LCD is 4x. Multiply through the entire equation by 4x to clear the fractions.
Distribute 4x to each term in the equation. This gives: 4x * (2/x) + 4x * (1/2) = 4x * (3/4). Simplify each term by canceling out the denominators where possible.
Simplify the resulting equation. After canceling the denominators, you should have: 8 + 2x = 3x.
Rearrange the equation to isolate the variable x. Subtract 2x from both sides to get: 8 = x.
Classify the equation. Since the equation has a single solution (x = 8), it is a conditional equation. A conditional equation is true only for specific values of the variable.
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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 any given equation.
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Solving Rational Equations
Rational equations involve fractions with variables in the denominator. To solve these equations, one typically finds a common denominator to eliminate the fractions, allowing for easier manipulation and simplification. This process is crucial for isolating the variable and determining the solution accurately.
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Checking Solutions
After solving an equation, it is essential to check the solution by substituting it back into the original equation. This verification process ensures that the solution is valid and helps identify the type of equation. It confirms whether the equation holds true for the found value, thus determining if it is an identity, conditional, or inconsistent.
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Related Practice
Textbook Question
Solve each absolute value inequality. - 4|1 - x| < - 16
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Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
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Textbook Question
Solve each polynomial equation in Exercises 86–87. 2x^4 = 50 x^2
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. 3 ≤ |2x - 1|
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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).]
a)b)c)d)e)f)
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