Algebra I to AP Precalculus · Module 1 of 14
Foundations: Expressions & Properties
Order of operations, the properties that justify every algebraic move, and translating words into expressions.
Order of Operations & Evaluating Expressions
- PEMDAS: Parentheses/grouping, Exponents, Multiplication & Division (left to right), Addition & Subtraction (left to right).
- To evaluate an expression, substitute the given value for each variable first, then apply order of operations.
- Multiplication and division share one priority level; addition and subtraction share another — within a level, work left to right.
Evaluate 3 + 2(x − 1)² when x = 4.
Solution. Substitute x = 4: 3 + 2(4 − 1)² = 3 + 2(3)². Exponent first: 3 + 2(9) = 3 + 18. Add: 21.
Common mistake: The exponent applies only to what's inside its own parentheses — not to the 2 multiplying it.
Properties of Real Numbers
- Commutative: a + b = b + a and ab = ba (order doesn't matter for + and ×, but does for − and ÷).
- Associative: (a + b) + c = a + (b + c) and (ab)c = a(bc) — grouping doesn't matter.
- Distributive: a(b + c) = ab + ac — the main tool for removing parentheses.
- Identity & inverse: a + 0 = a, a·1 = a; a + (−a) = 0, a·(1/a) = 1 for a ≠ 0.
Use the distributive property to expand −3(2x − 5).
Solution. Multiply −3 by each term inside: (−3)(2x) + (−3)(−5) = −6x + 15.
Common mistake: A negative sign in front of parentheses distributes to every term inside — dropping the sign on the second term is the single most common Algebra 1 slip.
Combining Like Terms & Simplifying
- Like terms have the exact same variable(s) raised to the exact same power(s); only their coefficients combine.
- Simplify by distributing first (if needed), then adding or subtracting the coefficients of like terms.
- Plain numbers (constants) are like terms with each other, separate from any variable terms.
Simplify 4x + 3(2x − 1) − 5.
Solution. Distribute: 4x + 6x − 3 − 5. Combine like terms: (4x + 6x) + (−3 − 5) = 10x − 8.
Translating Words into Expressions
- "Sum," "more than," "increased by" → addition. "Difference," "decreased by" → subtraction.
- "Product," "times," "of" → multiplication. "Quotient," "per," "divided by" → division.
- Assign a variable to the unknown first, then build the expression phrase by phrase, working from the end of the phrase for "less than.".
Translate "seven less than twice a number" into an algebraic expression.
Solution. "Twice a number" is 2n. "Seven less than" that is 2n − 7 — not 7 − 2n.
Common mistake: "Less than" reverses the order of the words — always translate it from the end of the phrase backward.
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. Evaluate 2(x + 3)² − 4 when x = 1.
2. Which property is shown by 5(x + 2) = 5x + 10?