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Questions and Answers
Algebraic manipulation involves rearranging and simplifying algebraic expressions using various techniques to solve ______, inequalities, and functions.
Algebraic manipulation involves rearranging and simplifying algebraic expressions using various techniques to solve ______, inequalities, and functions.
equations
The ______ Law of Addition states that the order of numbers being added does not change the result.
The ______ Law of Addition states that the order of numbers being added does not change the result.
Commutative
The ______ Law allows us to multiply a single value to multiple terms within parentheses.
The ______ Law allows us to multiply a single value to multiple terms within parentheses.
Distributive
Combining ______ Terms involves adding or subtracting terms with the same variable(s) and coefficient(s).
Combining ______ Terms involves adding or subtracting terms with the same variable(s) and coefficient(s).
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To remove parentheses, we use the ______ Law to expand expressions.
To remove parentheses, we use the ______ Law to expand expressions.
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The ______ Property allows us to add or subtract the same value to both sides of an equation to isolate the variable.
The ______ Property allows us to add or subtract the same value to both sides of an equation to isolate the variable.
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The ______ of a function refers to the set of input values.
The ______ of a function refers to the set of input values.
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When solving inequalities, we must ______ the inequality sign when multiplying or dividing both sides by a negative value.
When solving inequalities, we must ______ the inequality sign when multiplying or dividing both sides by a negative value.
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The ______ of Functions involves combining two or more functions by substitution.
The ______ of Functions involves combining two or more functions by substitution.
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Algebraic manipulation techniques are essential in solving various mathematical ______ and equations.
Algebraic manipulation techniques are essential in solving various mathematical ______ and equations.
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Study Notes
Algebraic Manipulation
Definition
Algebraic manipulation involves rearranging and simplifying algebraic expressions using various techniques to solve equations, inequalities, and functions.
Laws of Algebra
-
Commutative Law of Addition: The order of numbers being added does not change the result.
a + b = b + a -
Associative Law of Addition: The order in which numbers are added does not change the result.
(a + b) + c = a + (b + c) -
Distributive Law: Multiplying a single value to multiple terms within parentheses.
a(b + c) = ab + ac
Simplifying Expressions
- Combining Like Terms: Adding or subtracting terms with the same variable(s) and coefficient(s).
- Removing Parentheses: Using the distributive law to expand expressions with parentheses.
- Canceling Out Terms: Eliminating terms that are equal but opposite in sign.
Solving Equations
- Addition/Subtraction Property: Adding or subtracting the same value to both sides of an equation to isolate the variable.
- Multiplication/Division Property: Multiplying or dividing both sides of an equation by the same non-zero value to isolate the variable.
- Inverse Operations: Using opposite operations to isolate the variable (e.g., adding the opposite of a number to get rid of it).
Solving Inequalities
- Reversing the Inequality Sign: When multiplying or dividing both sides of an inequality by a negative value, the inequality sign is reversed.
Functions
- Domain and Range: The set of input values (domain) and the set of possible output values (range) of a function.
- Composition of Functions: Combining two or more functions by substitution.
These algebraic manipulation techniques are essential in solving various mathematical problems and equations.
Algebraic Manipulation
Laws of Algebra
- Algebraic manipulation involves rearranging and simplifying algebraic expressions using various techniques.
- The Commutative Law of Addition states that the order of numbers being added does not change the result, e.g.,
a + b = b + a. - The Associative Law of Addition states that the order in which numbers are added does not change the result, e.g.,
(a + b) + c = a + (b + c). - The Distributive Law allows multiplying a single value to multiple terms within parentheses, e.g.,
a(b + c) = ab + ac.
Simplifying Expressions
- Combining Like Terms involves adding or subtracting terms with the same variable(s) and coefficient(s).
- Removing Parentheses involves using the distributive law to expand expressions with parentheses.
- Canceling Out Terms involves eliminating terms that are equal but opposite in sign.
Solving Equations
- The Addition/Subtraction Property allows adding or subtracting the same value to both sides of an equation to isolate the variable.
- The Multiplication/Division Property allows multiplying or dividing both sides of an equation by the same non-zero value to isolate the variable.
- Inverse Operations involve using opposite operations to isolate the variable, e.g., adding the opposite of a number to get rid of it.
Solving Inequalities
- When multiplying or dividing both sides of an inequality by a negative value, the Inequality Sign is reversed.
Functions
- A function has a Domain (set of input values) and a Range (set of possible output values).
- Composition of Functions involves combining two or more functions by substitution.
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Description
Test your understanding of algebraic manipulation techniques and laws, including commutative, associative, and distributive laws, to solve equations and functions.
