Domanda del libro di testo
Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - (x - 3)/2 = (x + 2)/3
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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 41
Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 4/x = 5/2x + 3
Guida verificata passo dopo passo1
Identify the denominators in the equation \(\frac{4}{x} = \frac{5}{2x + 3}\). The denominators are \(x\) and \(2x + 3\).
Find the values of \(x\) that make the denominators zero to determine the restrictions. Solve \(x = 0\) and \(2x + 3 = 0\) separately.
From \(2x + 3 = 0\), solve for \(x\) by isolating \(x\): \(2x = -3\), so \(x = -\frac{3}{2}\). These values (\(x = 0\) and \(x = -\frac{3}{2}\)) are restrictions and cannot be solutions.
Multiply both sides of the original equation by the least common denominator (LCD), which is \(x(2x + 3)\), to eliminate the denominators. This gives: \(4(2x + 3) = 5x\).
Expand and simplify the resulting equation, then solve for \(x\). After finding the solutions, check each against the restrictions to ensure they are valid.
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Rational Equations
Rational equations are equations that involve fractions with polynomials in the numerator and denominator. Solving them often requires finding a common denominator or multiplying both sides by the least common denominator to eliminate fractions and simplify the equation.
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05:56
Introduction to Rational Equations
Restrictions on the Variable
Restrictions are values of the variable that make any denominator zero, which are undefined in mathematics. Identifying these values is crucial before solving the equation to avoid invalid solutions and ensure the solution set is correct.
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05:28Equations with Two Variables
Solving Linear Equations After Clearing Denominators
Once denominators are cleared by multiplying both sides by the least common denominator, the equation becomes a linear equation. Solving this involves isolating the variable using inverse operations like addition, subtraction, multiplication, or division.
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04:02Solving Linear Equations with Fractions
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