College Algebra - Week 1

Expressions and Equations

Every algebraic manipulation begins with expressions and equations. This week you build the foundation: evaluating expressions, simplifying using the order of operations, applying the properties of real numbers, and solving your first linear equations. Mastery here accelerates every topic that follows.

Learning Objectives

- Identify and classify real numbers (natural, integer, rational, irrational). - Apply the order of operations (PEMDAS) correctly to numerical and algebraic expressions. - Evaluate algebraic expressions by substituting given values. - Simplify expressions by combining like terms. - Apply the distributive property to expand and factor simple expressions. - Solve one-step and two-step linear equations in one variable. - Check a solution by substituting it back into the original equation.

1. The Real Number System

Real numbers include all numbers on the number line. - Natural numbers: 1, 2, 3, ... (counting numbers) - Whole numbers: 0, 1, 2, 3, ... - Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... - Rational numbers: any number expressible as p/q where p and q are integers and q is not 0. Examples: 1/2, -3/4, 0.75, 5 - Irrational numbers: cannot be expressed as a fraction of integers. Examples: sqrt(2), pi, e Every rational or irrational number is a real number. The real line is continuous and complete.

2. Order of Operations (PEMDAS)

Always evaluate expressions in this order: 1. Parentheses (and all grouping symbols) 2. Exponents 3. Multiplication and Division (left to right) 4. Addition and Subtraction (left to right) Example 1: Evaluate 3 + 4 * 2 Multiplication first: 4 * 2 = 8 Then addition: 3 + 8 = 11 Example 2: Evaluate (3 + 4) * 2 Parentheses first: 3 + 4 = 7 Then multiplication: 7 * 2 = 14 Example 3: Evaluate 2^3 + 4 / 2 - 1 Exponent: 2^3 = 8 Division: 4 / 2 = 2 Left to right: 8 + 2 - 1 = 9

3. Evaluating Algebraic Expressions

To evaluate an expression, replace each variable with its given value and compute. Example 1: Evaluate 3x - 5 when x = 4. 3(4) - 5 = 12 - 5 = 7 Example 2: Evaluate 2a^2 + b when a = 3 and b = -1. 2(3)^2 + (-1) = 2(9) - 1 = 18 - 1 = 17 Example 3: Evaluate (x + y)^2 - xy when x = 2 and y = -3. (2 + (-3))^2 - (2)(-3) = (-1)^2 - (-6) = 1 + 6 = 7

4. Properties of Real Numbers

These properties hold for all real numbers a, b, c: Commutative: a + b = b + a and a * b = b * a Associative: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c) Distributive: a(b + c) = ab + ac Identity: a + 0 = a and a * 1 = a Inverse: a + (-a) = 0 and a * (1/a) = 1 (for a not 0) Zero property: a * 0 = 0 These are not arbitrary rules — they are provable from the axioms of arithmetic and underlie every algebraic step you will take.

5. Combining Like Terms

Like terms have the same variable raised to the same power. 3x and -7x are like terms. 5x^2 and 2x are NOT like terms. To simplify, add or subtract the coefficients of like terms. Example 1: Simplify 4x + 3y - 2x + 5y = (4x - 2x) + (3y + 5y) = 2x + 8y Example 2: Simplify 3a^2 - 5a + 2a^2 + 7a - 1 = (3a^2 + 2a^2) + (-5a + 7a) - 1 = 5a^2 + 2a - 1

6. The Distributive Property

a(b + c) = ab + ac for any real numbers a, b, c. Example 1: Expand 3(x + 4) = 3x + 12 Example 2: Expand -2(5x - 3) = -10x + 6 Example 3: Expand 4(2x + 3y - 1) = 8x + 12y - 4 Example 4: Simplify 2(x + 3) + 5(x - 1) = 2x + 6 + 5x - 5 = 7x + 1 Warning: When distributing a negative sign, every term inside the parentheses changes sign. -(x - 4) = -x + 4

7. Introduction to Equations

An equation is a statement that two expressions are equal. 3x + 1 = 10 is an equation. A solution is a value that makes the equation true. To solve an equation, isolate the variable by performing the same operation on both sides. The three tools: 1. Add/subtract the same number to both sides. 2. Multiply/divide both sides by the same non-zero number. 3. Simplify each side first (distribute, combine like terms).

8. Solving One-Step Equations

Example 1: Solve x + 7 = 12. Subtract 7 from both sides: x = 5 Check: 5 + 7 = 12. True. Example 2: Solve 4x = 28. Divide both sides by 4: x = 7 Check: 4(7) = 28. True. Example 3: Solve x/3 = -5. Multiply both sides by 3: x = -15 Check: -15/3 = -5. True.

9. Solving Two-Step Equations

Example 1: Solve 3x + 4 = 19. Subtract 4: 3x = 15. Divide by 3: x = 5. Check: 3(5) + 4 = 15 + 4 = 19. True. Example 2: Solve 2x - 7 = -1. Add 7: 2x = 6. Divide by 2: x = 3. Example 3: Solve 5 - 3x = 20. Subtract 5: -3x = 15. Divide by -3: x = -5. Check: 5 - 3(-5) = 5 + 15 = 20. True.

10. Always Check Your Solution

After solving, substitute the answer back into the original equation. If the left side equals the right side, the solution is correct. If not, an error was made — go back and find it. Checking takes 30 seconds and prevents losing full credit on exams. This habit becomes especially important in later topics (rational equations, radical equations) where extraneous solutions can appear that seem to work algebraically but actually fail the original equation.

11. Word Problem Setup

Many algebra problems describe a situation in words. The skill is translating words into equations. Key word translations: - 'Is', 'equals', 'is the same as' → = - 'Sum of x and 5' → x + 5 - 'Difference of x and 3' → x - 3 - 'Twice a number' → 2x - 'A number increased by 7' → x + 7 Example: The sum of a number and 9 is 22. Find the number. Translate: x + 9 = 22. Solve: x = 13. Check: 13 + 9 = 22. True.

12. Common Mistakes

- Forgetting to apply the distributive property to all terms. - Dividing by a number instead of using the inverse operation (e.g., subtracting a product). - Sign errors when subtracting negative numbers: -(-3) = +3. - Treating 0/0 as valid in an equation step. - Combining unlike terms: 3x + 2 cannot be simplified to 5x.