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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.

−3 ∙

𝑥 = 11 ∙ −3 −3 𝑥 = −33

11x = 99 11𝑥 99 = 11 11 X=9 3+7+2=2+7+3 3∙7∙2=2∙3∙7

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.

Multiplicative Property of Zero

The product of any number and zero is zero.

Reflexive Property

All numbers are equal to themselves.

𝑥 = 11 ∙ −3 −3 𝑥 = −33 12 + 0 = 12 −3 ∙

-7 + 0 = -7 12 + (-12) = 0 -7 + 7 = 0 12 ∙ 1 = 12 -7,123 ∙ 1 = -7,123 1

12 ∙(12) = 1 −2 −5 ∙ =1 5 2 -12,345,678 ∙ 0 = 0

12 = 12 -4 = -4

Symmetric Property

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.

Substitution Property

-7 = x x = -7

Notes Algebraic Properties - filled in.pdf

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.

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