Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 139

Chapter 5, Problem 139

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

Verified step by step guidance

Start with the given equation:

\frac{x + 2}{4 x + 3} = \frac{1}{x}

To eliminate the fractions, cross-multiply both sides: multiply the numerator of the left fraction by the denominator of the right fraction and set it equal to the numerator of the right fraction times the denominator of the left fraction. This gives:

(x + 2) 	imes x = 1 	imes (4x + 3)

Expand both sides: on the left, distribute

x

over

(x + 2)

to get

x^2 + 2x

; on the right, you have

4x + 3

Set up the equation:

x^2 + 2x = 4x + 3

. Then, bring all terms to one side to set the equation equal to zero:

x^2 + 2x - 4x - 3 = 0

, which simplifies to

x^2 - 2x - 3 = 0

Solve the quadratic equation

x^2 - 2x - 3 = 0

by factoring, completing the square, or using the quadratic formula to find the values of

x

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Rational Equations

A rational equation involves fractions with polynomials in the numerator and denominator. To solve, find a common denominator or cross-multiply to eliminate fractions, then solve the resulting polynomial equation. Always check for values that make denominators zero, as these are excluded from the solution.

Cross-Multiplication

Cross-multiplication is a method used to solve equations where two fractions are set equal. Multiply the numerator of one fraction by the denominator of the other and set the products equal. This transforms the equation into a simpler polynomial form, making it easier to solve.

Domain Restrictions in Rational Expressions

The domain of a rational expression excludes values that make any denominator zero, as division by zero is undefined. When solving rational equations, identify and exclude these values from the solution set to ensure valid answers.

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Restrictions on Rational Equations

Related Practice

Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve for x: a(x - 2) = b(2x + 3)

Textbook Question

Without using a calculator, find the exact value of: [log₃ 81 - log_𝝅 1]/[log_2√2 8 - log 0.001]

Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Textbook Question

If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

Textbook Question

Without using a calculator, find the exact value of log₄ [log₃ (log₂ 8)].

Textbook Question

145. Without using a calculator, determine which is the greater number: log₄ 60 or log₃ 40.