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 137

Chapter 5, Problem 137

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

Verified step by step guidance

Start with the given equation:

a (x - 2) = b (2 x + 3)

Apply the distributive property to both sides to eliminate the parentheses:

a x - 2 a = 2 b x + 3 b

Group all terms containing

x

on one side and constant terms on the other side. For example, subtract

2 b x

from both sides and add

2 a

to both sides:

a x - 2 b x = 3 b + 2 a

Factor out

x

on the left side:

x (a - 2 b) = 3 b + 2 a

Finally, solve for

x

by dividing both sides by

a - 2 b

, assuming it is not zero:

x = \frac{3 b + 2 a}{a - 2 b}

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Key Concepts

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

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, a(x - 2) becomes ax - 2a. This step is essential to simplify both sides of the equation before solving for x.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. After simplifying both sides, combine like terms and use inverse operations such as addition, subtraction, multiplication, or division to solve for x.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable raised to the same power. This simplifies the equation and makes it easier to isolate the variable and solve the equation.

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Combinations

Related Practice

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

log_b (xy)⁵ = (log_b x + log_b y)⁵

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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

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

Textbook Question

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

Textbook Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

$l o g_{b} (x^{3} + y^{3}) = 3 l o g_{b} x + 3 l o g_{b} y$