College Algebra Study Guide: Fundamental Concepts and Problem Solving

Algebraic Expressions and Equations

Translating Verbal Statements to Algebraic Expressions

Translating word problems into algebraic expressions is a foundational skill in algebra. It involves identifying keywords and mathematical relationships.

• Key Point: "Four more than eighteen is less than the sum of 5 and x" translates to $18 + 4 < 5 + x$.

• Example: "Twice a number increased by seven" becomes $2x + 7$.

Prime Factorization

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

• Definition: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Example: The prime factorization of 280 is $280 = 2^3 \times 5 \times 7$.

Listing Factors

Factors of a number are integers that divide the number without leaving a remainder.

• Example: Factors of 32: 1, 2, 4, 8, 16, 32.

Solving Inequalities

Inequalities compare two expressions using symbols such as <, >, or =.

• Example: For $|5| - 2$ and $4 + (-3)$, determine which symbol ($<, >, =$) makes the statement true.

Fractions and Decimals

Expressing in Lowest Terms

Reducing fractions to lowest terms means dividing the numerator and denominator by their greatest common divisor.

• Example: $\frac{30}{45} = \frac{2}{3}$

Improper Fractions and Mixed Numbers

Improper fractions have numerators greater than denominators. Mixed numbers combine a whole number and a fraction.

• Example: $\frac{9}{4} = 2\frac{1}{4}$

Operations with Fractions

Adding, subtracting, multiplying, and dividing fractions require common denominators for addition/subtraction.

• Example: $\frac{2}{3} - \frac{1}{6} = \frac{1}{2}$

Decimals and Percents

Decimals and percents are alternative representations of fractions.

• Conversion: To convert a fraction to a decimal, divide the numerator by the denominator.

• Example: $\frac{1}{8} = 0.125$

• Percent: $2.75$ as a percent is $2.75 \times 100 = 275\%$

Rounding and Estimation

Rounding Numbers

Rounding is approximating a number to a specified place value.

• Example: $289.2749$ rounded to the nearest hundredth is $289.27$.

• Example: $381.009$ rounded to the nearest hundred is $400$.

Evaluating and Simplifying Expressions

Substitution and Evaluation

Substitute given values for variables and perform arithmetic operations.

• Example: Evaluate $2 + 5x = 3y$ when $x = 30$ and $y = -4$.

Simplifying Algebraic Expressions

Simplifying involves combining like terms and performing operations according to the order of operations (PEMDAS).

• Example: $7^2$ means $7 \times 7 = 49$.

• Example: $3(-8) = -24$.

Solving Equations

Linear Equations

Solving for a variable involves isolating it on one side of the equation.

• Example: Which numbers are solutions of $5x - 3 = x + 5$? Substitute $x = 1, 2, 3$ to check which values satisfy the equation.

Tables

Comparison of Fraction, Decimal, and Percent Forms

This table summarizes conversions between fractions, decimals, and percents.

Fraction	Decimal	Percent
$\frac{1}{8}$	0.125	12.5%
$\frac{3}{5}$	0.6	60%
$\frac{2}{5}$	0.4	40%
$\frac{7}{8}$	0.875	87.5%

Additional info:

• Some questions involve order of operations, absolute value, and evaluating expressions with negative numbers.

• Problems cover basic algebraic manipulation, suitable for College Algebra review.