Domanda del libro di testo
Find all values of x satisfying the given conditions. y1 = (2x - 1)/(x2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 67
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|3x - 2| = 14
Guida verificata passo dopo passo1
Start with the given equation: \(2|3x - 2| = 14\).
Isolate the absolute value expression by dividing both sides of the equation by 2: \(|3x - 2| = \frac{14}{2}\).
Simplify the right side: \(|3x - 2| = 7\).
Recall that if \(|A| = B\), where \(B > 0\), then \(A = B\) or \(A = -B\). So, set up two equations: \(3x - 2 = 7\) and \(3x - 2 = -7\).
Solve each equation separately for \(x\):
- For \(3x - 2 = 7\), add 2 to both sides and then divide by 3.
- For \(3x - 2 = -7\), add 2 to both sides and then divide by 3.
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Absolute Value Definition
The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A| = B, where B ≥ 0, the solutions satisfy A = B or A = -B. Understanding this is essential to split the equation into two cases for solving.
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08:07
Vertex Form
Solving Linear Equations
After isolating the absolute value expression, solving the resulting linear equations involves applying inverse operations like addition, subtraction, multiplication, and division. This process helps find the values of the variable that satisfy each case derived from the absolute value.
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04:02Solving Linear Equations with Fractions
Checking for No Solution
An absolute value equation has no solution if the expression inside the absolute value equals a negative number, since absolute values cannot be negative. It is important to verify the solutions and confirm that the original equation is satisfied.
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05:21Restrictions on Rational Equations
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