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NOTES: Algebraic Properties
ALGEBRAIC PROPERTY
VERBAL EXPLANATION
MODEL
Addition Property of Equality/Inequality
Adding the same amount to both sides of an equation maintains the equality of both sides. The same goes for inequalities
4x – 7 = 5 – 3x + 7 +7 4x = 12 – 3x
Subtraction Property of Equality/Inequality
Subtracting the same amount to both sides of an equation maintains the equality of both sides. The same goes for inequalities
4x + 1 = 5 – 3x -4x -4x 1 = 5 – 7x
Multiplication Property of Equality/Inequality
Multiplying by the same amount on both sides of an equation maintains the equality of both sides. The same goes for inequalities if the amounts are positive. If they are negative the inequality switches direction Dividing by the same amount on both sides of an equation maintains the equality of both sides. The same goes for inequalities if the amounts are positive. If they are negative the inequality switches direction
Division Property of Equality/Inequality
Commutative Property
Associative Property
Distributive Property
When adding/multiplying multiple numbers together, you may add/multiply in any order you choose.
When adding or multiplying groups of numbers, you may group the numbers together how you choose.
(3 + 7) + 2 = 3 + (7 + 2)
This property allows you to remove parenthesis around a group of terms by multiplying all terms inside the parenthesis by the factor outside the parenthesis
-4(12x + 8) = -48x – 32
3 ∙ (7 ∙ 2) = (3 ∙ 7) ∙ 2
Combining Like Terms
You may add/sub terms that contain the same variable factors with the same exponents
5 – 3x + 9x + 2 – x = 5 + 2 + 5x
Simplifying
This property allows you to perform the indicated operations on either side of the equation (usually add/sub/mult/div)
5 – 3x + 9x + 2 – x = 7 – 3x + 9x – x
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
The additive identity is zero, because when you add zero to any number, the number stays the same. The additive inverse of any number is the number multiplied by -1. The sum of any number and its additive inverse is zero. The multiplicative identity is one, because any number multiplied by one remains the same number. The multiplicative inverse of any number is its reciprocal. The product of any number and its inverse is one.
Changing which side is “right” or “left” of an equation does not make the equation untrue.
Transitive Property
Given three terms, if the 1st the 2nd A = B, B = -5, so A = -5 one are equal and the 2nd and 3rd one are equal, then the 1st and 3rd terms must be equal. If it is known that one term is y = 2x and 5x – y = 12. equal to a second one and the first So, 5x – 2x = 12 too. term exists in a new equation, then the second term may replace the first in the new equation.
goes for inequalities. 4x â 7 = 5 â 3x. + 7 +7. 4x = 12 â 3x. Subtraction Property of. Equality/Inequality. Subtracting the same amount to. both sides of an equation.
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