Questions and Answers
What is the result of the expression $3(2 + 4) - 5$?
Which of the following pairs of terms are like terms?
How would you isolate the variable in the equation $5x - 7 = 3$?
Which property of operations allows you to rearrange the addition of numbers, such as $a + b = b + a$?
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What is the inequality solution of $x + 4 < 10$?
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In the expression $2(x + 3) + 4$, what is the result after applying the distributive property?
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Study Notes
Integers and Operations
- Integers: Whole numbers that can be positive, negative, or zero.
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Operations:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
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Properties:
- Commutative Property: a + b = b + a; ab = ba
- Associative Property: (a + b) + c = a + (b + c); (ab)c = a(bc)
- Distributive Property: a(b + c) = ab + ac
- Order of Operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - PEMDAS.
Variables and Expressions
- Variables: Symbols (often letters) used to represent unknown values.
- Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 2).
- Evaluating Expressions: Substitute values for variables and perform operations.
- Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and 5x).
Equations and Inequalities
- Equations: Mathematical statements that two expressions are equal (e.g., 2x + 3 = 7).
- Solving Equations: Isolate the variable on one side (e.g., x = (7 - 3)/2).
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Inequalities: Statements that compare expressions (e.g., x + 2 > 5).
- Types: Greater than (>), Less than (<), Greater than or equal to (≥), Less than or equal to (≤).
- Graphing Inequalities: Use number lines or coordinate planes to represent solutions.
Ratios and Proportions
- Ratios: A comparison of two quantities (e.g., 3:5 means 3 parts of one quantity for every 5 parts of another).
- Proportions: An equation stating that two ratios are equal (e.g., a/b = c/d).
- Cross Multiplication: Used to solve proportions; if a/b = c/d, then ad = bc.
- Simplifying Ratios: Reduce to the simplest form by dividing both parts by their greatest common factor.
Functions and Graphs
- Function: A relation where each input has exactly one output (e.g., f(x) = 2x + 3).
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Domain and Range:
- Domain: All possible input values (x-values).
- Range: All possible output values (y-values).
- Graphing Functions: Plot points (x, f(x)) on a coordinate plane.
- Linear Functions: Forms a straight line when graphed; can be expressed as y = mx + b, where m is the slope and b is the y-intercept.
- Slope: The rate of change; calculated as (change in y)/(change in x).
Integers and Operations
- Integers encompass whole numbers that can be positive, negative, or zero.
- Four fundamental operations include:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)
- Properties of operations:
- Commutative Property: The order of addition or multiplication does not affect the result (a + b = b + a; ab = ba).
- Associative Property: The way numbers are grouped does not affect the sum or product ((a + b) + c = a + (b + c); (ab)c = a(bc)).
- Distributive Property: Multiplying a number by a sum is the same as doing each multiplication separately (a(b + c) = ab + ac).
- The Order of Operations is crucial for evaluation, represented by PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
Variables and Expressions
- Variables are symbols, typically letters, that signify unknown values.
- Expressions consist of numbers, variables, and operations (e.g., 3x + 2).
- Evaluating Expressions involves substituting specific values for variables and calculating the result.
- Like Terms are components that share the same variable raised to the same exponent (e.g., 3x and 5x can be combined).
Equations and Inequalities
- Equations are mathematical declarations indicating that two expressions are equivalent (e.g., 2x + 3 = 7).
- To Solve Equations, the goal is to isolate the variable on one side of the equation (e.g., x = (7 - 3)/2).
- Inequalities compare expressions and indicate relationships such as greater than (>), less than (<), and others.
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Description
Test your knowledge on integers and their operations including addition, subtraction, multiplication, and division. Explore important properties such as commutative, associative, and distributive. Challenge yourself with questions on the order of operations!
