Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces equations and inequalities, emphasizing the process of solving linear equations by transforming them into equivalent forms. Key techniques include isolating the variable, using properties of real numbers (such as adding, subtracting, multiplying, and dividing both sides by a nonzero quantity), and applying the Zero-Product Property when necessary. Applications to real-world problems, such as temperature conversion and investment scenarios, illustrate the practical use of algebra. Always remember to check solutions and consider domain restrictions.

Learning Objectives

1

Define key terms such as equation, solution, equivalent equation, identity, and linear equation.

2

Explain and apply procedures for manipulating and solving linear equations, including clearing fractions and the Zero-Product Property.

3

Determine the domain restrictions and verify solutions by substituting back into the original equation.

4

Model real-world problems using linear equations and solve for unknown quantities.

5

Utilize calculators to solve equations with fractions and round answers appropriately.

Key Concepts

CONCEPT

DEFINITION

Equation

A statement that two expressions, at least one of which contains a variable, are equal. It may be true or false depending on the variable's value.

Solution/Root

A value for the variable that, when substituted into the equation, makes the statement true.

Equivalent Equations

A sequence of equations obtained through legal algebraic manipulations that have the same solution set as the original equation.

Domain of the Variable

The set of all admissible values of the variable for which the expressions in the equation are defined.

Identity

An equation that is true for every value of the variable for which both sides are defined.

Zero-Product Property

If the product of two factors is zero, then at least one of the factors must be zero.

Linear Equation

An equation of the form ax + b = 0, where a and b are real numbers and a ≠ 0. It is also called a first-degree equation.

Example Problems

Example 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. The fact that $2(x+3)=2 x+6$ is because of the _____ Property. (pp. $9-13$ )

Example 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. The fact that $3 x=0$ implies that $x=0$ is a result of the _____ Property. (pp. $9-13$ )

Example 3

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. The domain of the variable in the expression $\frac{x}{x-4}$ is _____. (p. 21)

Example 4

True or False Multiplying both sides of an equation by any number results in an equivalent equation.

Example 5

An equation that is satisfied for every value of the variable for which both sides are defined is called a(n)_____ .
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Step-by-Step Explanations

QUESTION

Solve the linear equation 3x - 5 = 4.

STEP-BY-STEP ANSWER:

Step 1: Add 5 to both sides of the equation: 3x - 5 + 5 = 4 + 5, which simplifies to 3x = 9.
Step 2: Divide both sides by 3: 3x/3 = 9/3, resulting in x = 3.
Step 3: Check the solution by substituting x = 3 into the original equation: 3(3) - 5 = 9 - 5 = 4, which verifies the solution.
Final Answer: x = 3.

Solving a Simple Linear Equation

QUESTION

Solve the equation (1/2)x + 5/2 - 4 = (1/3)(12x - 12).

STEP-BY-STEP ANSWER:

Step 1: Identify the denominators (2 and 3) and multiply both sides by the least common multiple, which is 6.
Step 2: Multiply each term by 6 to eliminate fractions.
Step 3: Simplify the resulting equation and combine like terms.
Step 4: Isolate the variable x on one side and solve the resulting linear equation.
Step 5: Check the solution by substituting back into the original equation.
Final Answer: (The process yields a unique solution after simplification, e.g., x = -7 as shown in the example.)

Clearing Fractions in Equations

QUESTION

Solve the quadratic-like equation obtained from factoring: x² - 9 = 0.

STEP-BY-STEP ANSWER:

Step 1: Factor the equation as (x - 3)(x + 3) = 0.
Step 2: Set each factor equal to zero: x - 3 = 0 or x + 3 = 0.
Step 3: Solve for x: x = 3 or x = -3.
Final Answer: The solutions are x = 3 and x = -3.

Using the Zero-Product Property

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Common Mistakes

  • Failing to note domain restrictions when variables appear in denominators or under even roots.
  • Incorrectly applying the Zero-Product Property, especially when the equation is not properly factored.
  • Squaring both sides of an equation without recognizing that it may introduce extraneous solutions.
  • Errors in clearing fractions, such as multiplying by an incorrect least common multiple, leading to mistakes in simplification.
  • Not checking solutions in the original equation to ensure that no invalid (extraneous) solutions are included.