Domanda del libro di testo
In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.
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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 44
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. 7/2x - 5/3x = 22/3
Guida verificata passo dopo passo1
Identify the denominators in the equation \(\frac{7}{2x} - \frac{5}{3x} = \frac{22}{3}\). The denominators are \$2x$ and \$3x$.
Find the values of \(x\) that make any denominator zero. Set each denominator equal to zero: \(2x = 0\) and \(3x = 0\). Solve these to find the restrictions on \(x\).
Rewrite the equation to have a common denominator on the left side. The common denominator for \$2x$ and \$3x$ is \$6x$. Express each fraction with denominator \$6x$:
\(\frac{7}{2x} = \frac{7 \times 3}{6x} = \frac{21}{6x}\) and \(\frac{5}{3x} = \frac{5 \times 2}{6x} = \frac{10}{6x}\).
Combine the fractions on the left side: \(\frac{21}{6x} - \frac{10}{6x} = \frac{21 - 10}{6x} = \frac{11}{6x}\). So the equation becomes \(\frac{11}{6x} = \frac{22}{3}\).
Solve for \(x\) by cross-multiplying: \(11 \times 3 = 22 \times 6x\). Simplify and solve the resulting equation for \(x\), keeping in mind the restrictions found earlier.
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Rational Expressions and Denominators
Rational expressions are fractions that contain variables in the denominator. Understanding how to identify values that make denominators zero is crucial because these values are undefined and must be excluded from the solution set to avoid division by zero.
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Rationalizing Denominators
Finding Restrictions on the Variable
Restrictions are values of the variable that cause any denominator in the equation to be zero. Before solving, determine these values to ensure they are not included in the final solution, maintaining the equation's validity.
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05:28Equations with Two Variables
Solving Rational Equations
To solve rational equations, first clear denominators by multiplying both sides by the least common denominator (LCD). Then solve the resulting equation, and finally check solutions against restrictions to exclude any invalid answers.
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05:56Introduction to Rational Equations
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