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UnitReal Numbers & Properties

Core Concepts

PEMDAS — order of operations: Parentheses → Exponents → Multiply/Divide → Add/Subtract

  • Absolute value: \(|a|\) always returns a non-negative value.
  • \(|a| = b \Rightarrow a = b \text{ or } a = -b\) (when \(b \geq 0\))
  • Number sets: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
▸ Quick Example

\(2 + 3 \times 4^2 - 8 \div 2 = 2 + 3(16) - 4 = 2 + 48 - 4 = 46\)

Answer: 46
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Order of Operations ★★☆☆
What is the value of \(\ 3(2^2) + 4 \cdot 5 - 6 \div 2\ \)?
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Absolute Value ★★☆☆
Which values of \(x\) satisfy \(|2x - 3| = 7\)?
2
UnitAlgebraic Expressions

Core Concepts

Like terms — same variable(s) raised to the same power. Combine coefficients only.

Distributive Property: a(b + c) = ab + ac

  • Watch signs: \(-2(3x - 4) = -6x + 8\) (multiply each term)
  • Terms: coefficient × variable-part (e.g., \(5x^2\))
▸ Quick Example

\(4x + 2y - x + 3y = 3x + 5y\)

Combine like terms carefully.
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Combining Like Terms ★★☆☆
Simplify: \((5x^2 - 3x + 2) + (2x^2 + 7x - 5)\)
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Distributive Property ★★☆☆
Simplify: \(-2(3x - 4) + 5x\)
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UnitLinear Equations

Core Concepts

Solving one-variable equations: isolate \(x\) using inverse operations.

Literal equation: solve for one variable in terms of others.

  • Step 1: Distribute / clear parentheses
  • Step 2: Collect variable terms on one side
  • Step 3: Isolate the variable (divide or multiply)
▸ Quick Example

\(2(x+1) = x + 5 \Rightarrow 2x+2 = x+5 \Rightarrow x = 3\)

Answer: x = 3
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Solving Equations ★★★☆
Solve for \(x\): \(\ 3(x + 2) = 2x + 10\)
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Literal Equations ★★★☆
The formula for the area of a triangle is \(A = \dfrac{1}{2}bh\). Solve for \(h\).
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UnitInequalities

Core Concepts

Key rule: flip the inequality sign when multiplying/dividing by a negative

Compound inequalities: \(a \lt x \leq b\) — solve each part simultaneously.

  • \(−3 < 2x + 1 \leq 9\): subtract 1 throughout, then divide by 2
  • Graph solutions on a number line
▸ Quick Example

\(-2x > 6 \Rightarrow x < -3\) (flip sign when dividing by −2)

Answer: x < −3
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Compound Inequality ★★★☆
Solve the compound inequality: \(-3 \lt 2x + 1 \leq 9\)
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Inequality · Word Problem ★★★☆
For what values of \(x\) is \(5x + 3 \gt 2x + 15\) true?
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UnitGraphing Linear Equations

Core Concepts

Slope-intercept form: y = mx + b  (m = slope, b = y-intercept)

Slope formula: m = (y₂ − y₁) / (x₂ − x₁)

  • Convert standard form \(Ax + By = C\) → divide both sides by \(B\)
  • Parallel lines: equal slopes; Perpendicular: slopes are negative reciprocals
▸ Quick Example

\(6x - 3y = 12 \Rightarrow y = 2x - 4\), so slope = 2, y-int = −4

Answer: slope = 2
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Slope ★★☆☆
What is the slope of the line passing through \((1,\ 3)\) and \((4,\ 12)\)?
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y-Intercept ★★★☆
What is the \(y\)-intercept of the line \(4x - 2y = 8\)?
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UnitSystems of Equations

Core Concepts

Substitution: solve one equation for a variable, substitute into the other.

Elimination: add/subtract equations to cancel a variable.

  • If coefficient signs are opposite → add the equations
  • If same → subtract one from the other
  • Always substitute back to find the second variable
▸ Quick Example

\(y = x + 2,\quad 2x + y = 8 \Rightarrow 2x + (x+2) = 8 \Rightarrow x = 2, y = 4\)

Solution: (2, 4)
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Substitution ★★★☆
Solve the system: \(y = 2x - 1\) and \(3x + y = 14\).
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Elimination ★★★★
Solve by elimination:
\(2x + 3y = 12\)
\(4x - 3y = 6\)
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UnitExponents & Powers

Core Concepts

Laws of Exponents:

  • Product rule: xᵃ · xᵇ = xᵃ⁺ᵇ
  • Power rule: (xᵃ)ᵇ = xᵃᵇ
  • Negative exp: x⁻ⁿ = 1/xⁿ
  • Power of a product: (ab)ⁿ = aⁿbⁿ
▸ Quick Example

\((3x^2)^3 = 3^3 \cdot x^{2 \times 3} = 27x^6\)

Apply power rule to each factor.
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Power Rule ★★★☆
Simplify: \(\left(2x^3y^2\right)^3\)
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Negative Exponents ★★★★
Simplify \(\left(3x^{-2}\right)^2\) and write with a positive exponent.
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UnitPolynomials

Core Concepts

FOIL (multiply two binomials): First · Outer · Inner · Last

Factoring trinomials \(x^2 + bx + c\): find two numbers that multiply to c and add to b.

  • \((x+p)(x+q) = x^2 + (p+q)x + pq\)
  • Always check by expanding your answer
▸ Quick Example

\(x^2 + 5x + 6 = (x+2)(x+3)\) because \(2+3=5\) and \(2\times3=6\)

Answer: (x+2)(x+3)
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Multiplying Binomials ★★★☆
Expand: \((x + 3)(x - 5)\)
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Factoring Trinomials ★★★☆
Factor completely: \(x^2 + 7x + 12\)
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UnitQuadratic Equations

Core Concepts

Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a

Completing the Square: rewrite \(ax^2+bx+c=0\) in the form \((x-h)^2 = k\).

  • Discriminant \(\Delta = b^2 - 4ac\): if >0 → 2 real roots; =0 → 1; <0 → no real roots
  • Perfect square: \((x-3)^2 = 25 \Rightarrow x - 3 = \pm 5\)
▸ Quick Example

\(x^2 - x - 6 = 0 \Rightarrow (x-3)(x+2)=0 \Rightarrow x=3 \text{ or } x=-2\)

Always verify by substituting back.
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Quadratic Formula ★★★★
Using the quadratic formula, solve \(2x^2 - 5x - 3 = 0\).
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Completing the Square ★★★★
Solve for \(x\): \(x^2 - 6x + 9 = 25\)
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UnitFunctions

Core Concepts

Function notation: \(f(x)\) means "the output when the input is \(x\)."

Domain of \(\sqrt{g(x)}\): set \(g(x) \geq 0\) and solve.

  • To find \(f(a)\): substitute \(x = a\) everywhere in the formula
  • Vertical line test: a graph is a function if every vertical line hits it at most once
▸ Quick Example

\(f(x) = x^2 + 1,\quad f(3) = 9 + 1 = 10\)

Answer: 10
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Function Notation ★★★☆
If \(f(x) = x^2 - 3x + 2\), what is \(f(-1)\)?
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Domain of Function ★★★★
What is the domain of \(f(x) = \sqrt{2x - 6}\)?
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