Book cover for College Algebra: Real Mathematics, Real People

College Algebra: Real Mathematics, Real People

Ron Larson

ISBN #9781305778917

7th Edition

5,230 Questions

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38,634 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers fundamental concepts in algebra and number theory, including the classification of numbers, the use of interval notation for representing sets, and the identification of terms and coefficients in expressions. It also demonstrates key arithmetic properties and shows how absolute value is used in real-world applications such as budget variance tests. Mastering these basics prepares students for more advanced problem solving in both pure and applied mathematics.

Learning Objectives

1

Classify different types of numbers (rational, prime, irrational) and understand their properties.

2

Interpret and express sets using inequality notation and interval notation.

3

Identify and extract terms and coefficients from algebraic expressions.

4

Apply key arithmetic properties (distributive, commutative, associative) in simplifying expressions.

5

Analyze real-world problems such as budget variance tests using absolute value comparisons.

Key Concepts

CONCEPT

DEFINITION

Rational Number

A number that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0.

Prime Number

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Term

A single mathematical expression or component in an algebraic expression, usually separated by addition or subtraction.

Coefficient

A numerical factor that multiplies a variable in an algebraic term.

Distributive Property

A property stating that a(b + c) = ab + ac.

Commutative Property of Addition

A property stating that a + b = b + a.

Associative Property of Addition

A property stating that (a + b) + c = a + (b + c).

Absolute Value

The non-negative value of a number, denoted by |x|.

Interval Notation

A method of writing subsets of the real numbers using brackets and parentheses to denote closed or open endpoints.

Bounded/Unbounded

A set is bounded if it has both an upper and lower limit, and unbounded if it extends infinitely in one or both directions.

Example Problems

Example 1

Fill in the blank(s). A real number is _____ when it can be written as the ratio $\frac{p}{q}$ of two integers, where $q \neq 0.$

Example 2

Fill in the blank(s). ______ numbers have infinite non repeating decimal representations.

Example 3

Fill in the blank(s). A _____ number is an integer with exactly two positive factors: itself and $1 .$

Example 4

Fill in the blank(s). An algebraic expression is a combination of letters called _____ and real numbers called _____ .

Example 5

Fill in the blank(s). The ______ of an algebraic expression are those parts separated by addition.

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Step-by-Step Explanations

QUESTION

Express the set of all real numbers less than or equal to 5 in interval notation.

STEP-BY-STEP ANSWER:

Step 1: Understand that 'x ≤ 5' means all numbers x that are less than or equal to 5.
Step 2: Recognize that the lower bound is negative infinity since there is no minimum value specified.
Step 3: Use a closed bracket at 5 to indicate that 5 is included.
Final Answer: (-∞, 5]

Interval Notation (x ≤ 5)

QUESTION

In the expression 7x + 4, identify the terms and the coefficient of the variable term.

STEP-BY-STEP ANSWER:

Step 1: Identify the two parts separated by the plus sign: 7x and 4.
Step 2: Recognize 7x as the term that contains the variable x and 4 as the constant term.
Step 3: The coefficient for the term 7x is 7.
Final Answer: Terms are 7x and 4; the coefficient is 7.

Identifying Terms and Coefficients

QUESTION

Given actual expenses of $113,356 and budgeted expenses of $112,700 with a tolerance of $500, determine if the budget variance test is passed.

STEP-BY-STEP ANSWER:

Step 1: Calculate the absolute difference between the actual and budget amounts: |113,356 - 112,700| = 656.
Step 2: Compare the difference (656) with the tolerance ($500).
Step 3: Since 656 > 500, the variance is greater than the allowed tolerance.
Final Answer: The budget variance test is not met.

Budget Variance Test Using Absolute Value

QUESTION

Determine the absolute value of -3.

STEP-BY-STEP ANSWER:

Step 1: Recognize that absolute value represents the non-negative distance from zero.
Step 2: The absolute value of -3 is 3.
Final Answer: | -3 | = 3.

Absolute Value of an Expression

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Common Mistakes

  • Confusing rational and irrational numbers, such as mistakenly considering ?2 as rational.
  • Incorrect interval notation, for example, using a parenthesis instead of a bracket when the endpoint is included.
  • Misidentifying terms and coefficients in algebraic expressions, such as overlooking an implied coefficient of 1.
  • Errors in applying the distributive property, which can lead to incorrect expansion of expressions.
  • Calculation mistakes when evaluating absolute values or comparing actual differences against a tolerance.