Unit 1 · Expressions & Operations
Evaluate expressions in this order:
P → E → M/D (left to right) → A/S (left to right)
Key rule: Multiplication/Division have equal priority — work left to right. Same for Addition/Subtraction.
Quick Example
\( 2 + 3^2 \times 4 - 6 \div 2 \)
Step 1: \( 3^2 = 9 \) → Step 2: \( 9 \times 4 = 36 \) → Step 3: \( 6 \div 2 = 3 \) → Step 4: \( 2 + 36 - 3 = 35 \)
Answer: 35
Substitute the given value, then apply PEMDAS
Always use parentheses when substituting a negative number: if \(x = -3\), write \((-3)^2\), not \(-3^2\).
Quick Example
Evaluate \( 3x^2 + 2 \) when \( x = -2 \):
\( 3(-2)^2 + 2 = 3(4) + 2 = 12 + 2 = 14 \)
Answer: 14
Like terms: same variable AND same exponent
Distribute: a(b + c) = ab + ac
When distributing a negative, flip the sign of every term inside: \(-2(3x - 5) = -6x + 10\)
Quick Example
Simplify \( -3(2x - 4) + 5x \):
\( -6x + 12 + 5x = -x + 12 \)
Answer: −x + 12
Unit 2 · Equations & Inequalities
Goal: isolate the variable using inverse operations
Whatever you do to one side, do to the other
Check: substitute your answer back into the original equation.
Quick Example
Solve \( 5x - 3 = 17 \):
Add 3: \( 5x = 20 \) → Divide by 5: \( x = 4 \)
Answer: x = 4
Treat the target variable like a regular variable — isolate it using inverse operations
Example: Solve \( P = 2l + 2w \) for \( w \): subtract \( 2l \), then divide by 2: \( w = \dfrac{P - 2l}{2} \)
⚠ Multiply or divide by a NEGATIVE → FLIP the inequality sign
|ax + b| = c → ax + b = c OR ax + b = −c
For \( |x| = 5 \): \( x = 5 \) or \( x = -5 \). Always write two cases.
Quick Example
Solve \( -2x + 4 \leq 10 \):
Subtract 4: \( -2x \leq 6 \) → Divide by −2 (flip!): \( x \geq -3 \)
Answer: x ≥ −3
Unit 3 · Linear Functions & Graphs
Slope: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \) Slope-intercept: \( y = mx + b \)
Point-slope form: \( y - y_1 = m(x - x_1) \) — use when given slope and one point.
Quick Example
Find slope through \( (1, 2) \) and \( (3, 8) \):
\( m = \dfrac{8 - 2}{3 - 1} = \dfrac{6}{2} = 3 \)
Answer: m = 3
Unit 4 · Systems of Equations
Substitution: solve one equation for one variable, substitute into other
Elimination: add/subtract equations to cancel one variable
Always check your solution in BOTH original equations.
Quick Example — Substitution
Solve: \( y = 2x \) and \( x + y = 9 \):
Sub: \( x + 2x = 9 \Rightarrow 3x = 9 \Rightarrow x = 3,\; y = 6 \)
Answer: (3, 6)
Unit 5 · Exponents & Polynomials
\( x^a \cdot x^b = x^{a+b} \quad (x^a)^b = x^{ab} \quad (xy)^n = x^n y^n \)
FOIL for binomials: First, Outer, Inner, Last
Quick Example — FOIL
\( (x+2)(x-4) \):
First: \( x^2 \), Outer: \( -4x \), Inner: \( +2x \), Last: \( -8 \)
Result: \( x^2 - 2x - 8 \)
Answer: x² − 2x − 8
Unit 6 · Factoring
GCF first, always.
For \( x^2 + bx + c \): find two numbers that multiply to \(c\) and add to \(b\)
Check: expand your factored form to verify it matches the original.
Quick Example
Factor \( x^2 - 7x + 12 \):
Need: multiply to 12, add to −7 → \((-3) \times (-4) = 12\), \((-3)+(-4) = -7\)
\( (x - 3)(x - 4) \)
Answer: (x − 3)(x − 4)
Unit 7 · Functions
Domain: all valid inputs (x-values)
Range: all resulting outputs (y-values)
\( f(a) \) means "evaluate f at x = a"
For \( \sqrt{x - k} \): domain requires \( x - k \geq 0 \), so \( x \geq k \).
Quick Example
If \( f(x) = x^2 + 1 \), find \( f(-3) \):
\( f(-3) = (-3)^2 + 1 = 9 + 1 = 10 \)
Answer: 10
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