Chapter 1: Precalculus Preparation and Algebra Review

Why This Chapter Matters

Order of OperationsNested arithmetic with parentheses and powers. Distributive PropertyNegative factors and multi-term parentheses. Exponent RulesProducts, powers, quotients, and negative exponents. Systems of EquationsGraphing, substitution, and elimination. Simple InterestA classic linear model: I = Prt.

Most precalculus mistakes are not mysterious. They come from small algebra slips: distributing a negative sign incorrectly, combining unlike terms, losing a denominator, or applying exponent rules where they do not belong. This chapter reviews the tools we use constantly in functions, trigonometry, logarithms, and teacher certification math problems.

Core Concept

Before using a new formula, simplify the algebra structure: parentheses first, powers next, multiplication or division, then addition or subtraction.

Order of Operations

The order of operations tells us what to simplify first when an expression has several operations. A useful version is:

Parentheses \to Exponents \to Multiplication/Division \to Addition/Subtraction

Multiplication and division have the same priority, so work left to right. Addition and subtraction also have the same priority, so work left to right there too.

Worked Example

Simplify $3 + 2(5^2 - 9)$ .

3 + 2(25 - 9) = 3 + 2(16) = 3 + 32 = 35

Core Concept

Do the operation inside a grouping symbol before multiplying by the number outside it.

The Distributive Property

The distributive property lets one factor multiply every term inside parentheses.

a(b + c) = ab + ac

This rule is especially important when the outside factor is negative. The negative sign must distribute to every term.

Worked Example

Simplify $-3(2x - 5) + 4x$ .

-3(2x - 5) + 4x = -6x + 15 + 4x = -2x + 15

Core Concept

When distributing a negative number, every term inside the parentheses changes according to that negative factor.

Combining Like Terms

Like terms have the same variable part. We may combine $7x$ and $-2x$ , but we may not combine $7x$ and $7x^2$ .

4x^2 + 7x - 3x^2 + 5 - 2x = x^2 + 5x + 5

Core Concept

Combine coefficients only when the variable letters and exponents match exactly.

Exponent Rules

Exponent rules are shortcuts for repeated multiplication. They work when the bases match or when a power is applied to a product or quotient.

$x^a x^b = x^{a+b}$
$x^a / x^b = x^{a-b}$ , when $x \neq 0$
$(x^a)^b = x^{ab}$
$(xy)^a = x^a y^a$
$x^{-a} = 1 / x^a$ , when $x \neq 0$

Worked Example

Simplify $(2x^3y)^2$ .

(2x^3y)^2 = 2^2(x^3)^2y^2 = 4x^6y^2

Core Concept

A power outside parentheses applies to every factor inside, not just the variable that stands out first.

Fractions and Rational Expressions

Fractions behave predictably when we remember that a denominator divides the entire numerator. In algebra, restrictions matter: any value that makes a denominator zero is not allowed.

Worked Example

Simplify $(6x^2 - 9x) / 3x$ , with $x \neq 0$ .

\frac{6x^2 - 9x}{3x} = \frac{3x(2x - 3)}{3x} = 2x - 3

Core Concept

Cancel common factors, not pieces of terms. Factor first when a numerator or denominator has addition or subtraction.

Solving Linear Equations

Solving an equation means undoing operations while keeping both sides balanced. The goal is to isolate the variable.

Worked Example

Solve $4(2x - 1) = 3x + 11$ .

8x - 4 = 3x + 11 5x - 4 = 11 5x = 15 x = 3

Core Concept

Clear parentheses, gather variable terms on one side, gather number terms on the other, then divide by the coefficient.

Systems of Equations

A system asks for values that make more than one equation true at the same time. In graphing language, the solution is the intersection point.

Graphing IntersectionRead the shared point from two lines. SubstitutionReplace one variable expression inside the other equation. Add/Subtract EliminationCancel one variable by adding equations.

Core Concept

For systems, choose the method that cancels or replaces a variable fastest.

Worked Example 1.7.1 — Substitution Pair

Solve the system $y=2x+1$ and $x+y=10$ .

Substitute the expression for $y$ into the second equation.x+(2x+1)=10
Solve the resulting one-variable equation.3x+1=10 \rightarrow x=3
Substitute back to find $y$ .y=2(3)+1=7

The solution is $(3,7)$ .

Worked Example 1.7.2 — Elimination Pair

Solve the system $2x+y=11$ and $3x-y=9$ .

Add the equations so the $y$ terms cancel.(2x+y)+(3x-y)=11+9
Solve for $x$ .5x=20 \rightarrow x=4
Substitute back to find $y$ .2(4)+y=11 \rightarrow y=3

Simple Interest

Simple interest is a classic linear model. The interest grows by the same amount each year because interest is calculated only on the original principal.

I = Prt

Core Concept

The simple interest model multiplies principal, annual rate, and time.

Worked Example 1.8.1 — Savings Certificate

A savings certificate has a principal of $$800$ , an annual simple interest rate of $4%$ , and a time of $3$ years. Find the interest.

Translate the percent into a decimal rate.4% = 0.04
Substitute the principal, rate, and time into the model.I = Prt = 800(0.04)(3)
Multiply to find the interest earned.I = 96

The certificate earns $$96$ in simple interest.