College Algebra: Fundamental Concepts and Problem Solving

Order of Operations and Evaluating Expressions

Understanding Order of Operations

Order of operations is a set of rules that determines the correct sequence to evaluate a mathematical expression. This ensures consistency and accuracy in solving problems.

• PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted as $|x|$.

Example: Evaluate $6 + 4 \times | -3 |$

• First, calculate the absolute value: $| -3 | = 3$

• Then, multiplication: $4 \times 3 = 12$

• Finally, addition: $6 + 12 = 18$

Evaluating Algebraic Expressions

Substitution and Simplification

To evaluate an algebraic expression, substitute the given values for the variables and simplify using the order of operations.

• Substitution: Replace each variable with its given value.

• Simplification: Perform operations according to the order of operations.

Example: Evaluate $x^2 + 5x + 4$ for $x = -1$

• Substitute: $(-1)^2 + 5(-1) + 4$

• Calculate: $1 - 5 + 4 = 0$

Properties of Exponents

Exponent Rules

Exponents are used to represent repeated multiplication. Understanding their properties is essential for simplifying expressions.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Example: Simplify $(-3)^2$ and $-3^2$

• $(-3)^2 = (-3) \times (-3) = 9$

• $-3^2 = -(3^2) = -9$

Fractions and Rational Numbers

Reducing Fractions and Finding Common Denominators

Fractions can be simplified by dividing the numerator and denominator by their greatest common factor (GCF). To add or subtract fractions, a common denominator is required.

• Lowest Terms: A fraction is in lowest terms when the numerator and denominator have no common factors other than 1.

• Common Denominator: The least common denominator (LCD) is the smallest number that is a multiple of each denominator.

Example: Write $\frac{4}{3}$ with a denominator of 12.

• Find the equivalent fraction: $\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}$

Prime Factorization

Breaking Down Numbers into Prime Factors

Prime factorization is expressing a number as a product of its prime numbers.

• Prime Number: A number greater than 1 that has no positive divisors other than 1 and itself.

• Composite Number: A number that can be factored into smaller positive integers.

Example: Prime factorization of 45:

• $45 = 5 \times 9 = 5 \times 3 \times 3 = 5 \times 3^2$

Solving Linear Equations

Steps to Solve Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one.

• Isolate the variable: Use inverse operations to get the variable alone on one side of the equation.

• Balance the equation: Whatever you do to one side, do to the other.

Example: Solve $4x - 3 = 5$

• Add 3 to both sides: $4x = 8$

• Divide by 4: $x = 2$

Translating and Solving Word Problems

Setting Up Equations from Words

Word problems require translating English statements into mathematical equations.

• Identify variables: Assign a variable to the unknown quantity.

• Translate phrases: For example, "five less than a number" becomes $x - 5$.

• Set up and solve: Write the equation and solve for the variable.

Example: "The sum of twice a number and four is negative six."

• Let $x$ be the number. Equation: $2x + 4 = -6$

• Solve: $2x = -10 \implies x = -5$

Classification of Numbers

Rational and Irrational Numbers

Numbers can be classified based on their properties.

• Rational Numbers: Numbers that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., $\sqrt{2}$, $\pi$).

Example Table: Classification of Numbers

Number	Rational?
$\frac{8}{6}$	Yes
$0$	Yes
$\sqrt{13}$	No
$-3$	Yes
$\frac{11}{0}$	No (undefined)

Absolute Value

Definition and Properties

The absolute value of a number is its distance from zero, always non-negative.

• Notation: $|x|$

• Properties: $|a| \geq 0$ for any real number $a$; $|a| = a$ if $a \geq 0$, $|a| = -a$ if $a < 0$.

Example: $| -7 | = 7$

Summary Table: Key Properties and Examples

Concept	Definition	Example
Order of Operations	Sequence to evaluate expressions	$2 + 3 \times 4 = 2 + 12 = 14$
Absolute Value	Distance from zero	$| -5 | = 5$
Exponent Rules	Properties of exponents	$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
Prime Factorization	Product of primes	$24 = 2^3 \times 3$
Linear Equation	First-degree equation	$3x + 2 = 11 \implies x = 3$

Additional info: These notes cover foundational topics in College Algebra, including evaluating expressions, properties of exponents, fractions, solving equations, and translating word problems into algebraic equations. The content is structured to support exam preparation and mastery of core algebraic skills.