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Algebra Rules and Properties

Properties of numbers, rules for solving equations and inequalities, and common factoring patterns.

56 cardsGrades 6–8, Grades 9–12#algebra#properties#rules

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  • Commutative property of additionRule
    a + b = b + a
  • Commutative property of multiplicationRule
    ab = ba
  • Associative property of additionRule
    (a + b) + c = a + (b + c)
  • Associative property of multiplicationRule
    (ab)c = a(bc)
  • Distributive propertyRule
    a(b + c) = ab + ac
  • Additive identityRule
    a + 0 = a
  • Multiplicative identityRule
    a × 1 = a
  • Additive inverseRule
    a + (−a) = 0
  • Multiplicative inverseRule
    a × (1/a) = 1, for a not zero
  • Zero property of multiplicationRule
    a × 0 = 0
  • Reflexive propertyRule
    a = a
  • Symmetric propertyRule
    If a = b, then b = a
  • Transitive propertyRule
    If a = b and b = c, then a = c
  • Addition property of equalityRule
    If a = b, then a + c = b + c
  • Multiplication property of equalityRule
    If a = b, then ac = bc
  • Substitution propertyRule
    If a = b, then b can replace a in any expression
  • Rule for multiplying two negativesRule
    The product is positive
  • Rule for multiplying a negative and a positiveRule
    The product is negative
  • Rule for dividing two negativesRule
    The quotient is positive
  • Subtracting a negativeRule
    a − (−b) = a + b
  • Solve x + 7 = 12Rule
    x = 5
  • Solve 3x = 21Rule
    x = 7
  • Solve 2x − 5 = 9Rule
    x = 7
  • Solve x/4 = 6Rule
    x = 24
  • Solve 5x + 3 = 2x + 15Rule
    x = 4
  • Rule for inequalitiesRule
    Multiplying or dividing by a negative number flips the inequality sign
  • Solve −2x > 6Rule
    x < −3
  • Solve |x| = 5Rule
    x = 5 or x = −5
  • Solve |x| < 3Rule
    −3 < x < 3
  • Solve |x| > 3Rule
    x < −3 or x > 3
  • FOILRule
    First, Outer, Inner, Last: multiplying two binomials
  • Expand (x + 3)(x + 2)Rule
    x² + 5x + 6
  • Expand (x + 4)²Rule
    x² + 8x + 16
  • Expand (x − 5)²Rule
    x² − 10x + 25
  • Expand (x + 3)(x − 3)Rule
    x² − 9
  • Factor x² + 7x + 12Rule
    (x + 3)(x + 4)
  • Factor x² − 9Rule
    (x − 3)(x + 3)
  • Factor x² − 5x + 6Rule
    (x − 2)(x − 3)
  • Factor x² + 2x + 1Rule
    (x + 1)²
  • Factor 2x² + 6xRule
    2x(x + 3)
  • Sum of cubesRule
    a³ + b³ = (a + b)(a² − ab + b²)
  • Difference of cubesRule
    a³ − b³ = (a − b)(a² + ab + b²)
  • Binomial cubeRule
    (a + b)³ = a³ + 3a²b + 3ab² + b³
  • Solve x² = 49Rule
    x = 7 or x = −7
  • Solve x² − 5x + 6 = 0Rule
    x = 2 or x = 3
  • Midpoint of two numbersRule
    Their average: (a + b)/2
  • Cross-multiplicationRule
    If a/b = c/d, then ad = bc
  • Adding fractions with unlike denominatorsRule
    Use a common denominator first
  • Slope between two pointsRule
    (y₂ − y₁)/(x₂ − x₁)
  • Perpendicular to slope 2Rule
    Slope −1/2
  • Distance, rate, timeRule
    d = r × t
  • Percent changeRule
    (new − old) / old × 100%
  • Simple interestRule
    I = P r t
  • Average (mean) formulaRule
    Sum of values divided by number of values
  • Order of solving with variables on both sidesRule
    Collect variable terms on one side, constants on the other, then divide
  • Check a solutionRule
    Substitute it back into the original equation

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