Questions and Answers
What is a rational number?
What is an irrational number?
Any real number that cannot be expressed as a ratio of integers.
Define integers.
Whole numbers that include all the positives and negatives.
What is a whole number?
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What are natural numbers?
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What is an algebraic expression?
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What is a variable?
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What is a coefficient?
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What is an equation?
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Give an example of a solution to an equation with one variable.
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What are like terms?
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What is a literal equation?
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What is a proportion?
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What is an inequality?
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What does 'at least' mean?
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What does 'at most' mean?
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What is a compound inequality?
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Define relation in mathematical terms.
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What is a function?
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Define domain.
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Define range.
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What is an independent variable?
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What is a dependent variable?
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Provide an example of a linear function.
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What is the standard form of a linear equation?
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Define slope.
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What is the slope formula?
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What is the x-intercept?
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What is the y-intercept?
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What is slope-intercept form?
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What is point-slope form?
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What is an absolute value function?
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Define scatter plot.
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What is a solution of a system of linear equations?
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Provide an example of a system with one solution.
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What is a system with no solutions?
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What is a system with many solutions?
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What is an exponential function?
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What is an exponent?
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Define monomial.
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What is a binomial?
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What is a trinomial?
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Define polynomial.
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The rules of exponents govern the manipulation of _____ in expressions.
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What does multiplying powers entail?
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What is the power of power rule?
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What is the power of product rule?
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What is a negative exponent?
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Study Notes
Number Types
- Rational Number: Can be expressed as a ratio of integers.
- Irrational Number: Cannot be expressed as a ratio of integers.
- Integers: Whole numbers including negatives.
- Whole Numbers: Integers without negatives.
- Natural Numbers: Positive integers including zero.
Algebra Fundamentals
- Algebraic Expression: Combines integers, variables, and algebraic operations.
- Variable: Represents a constant or changing value, typically denoted as X or Y.
- Coefficient: A number multiplied by a variable in an expression.
- Equation: States that two mathematical expressions are equal.
Solving Equations
- Example of solution: For the equation (3x - 4 = 5), the objective is finding x.
- Like Terms: Terms sharing the same variable and exponent.
- Literal Equation: A formula where a specific variable is solved.
Ratios and Inequalities
- Proportion: Two ratios are equal, e.g., (1/2 = 5/10).
- Inequality: A relation indicating one expression is not equal to another (e.g., >, <).
- Compound Inequality: Combines two inequalities using "and" or "or."
Functions and Relations
- Relation: Set of ordered pairs from two different sets.
- Function: A relation mapping each input to exactly one output.
- Domain: Set of input values (X-axis).
- Range: Set of output values (Y-axis).
- Independent Variable: Input of a function.
- Dependent Variable: Output of a function.
Linear Equations & Functions
- Linear Equation (2 Variables): Example form (2x - 3y = 8).
- Linear Function: Standard form (y = mx + b), where m is the slope.
- Standard Form: Written as (ax + by = c).
Graph Characteristics
- Slope: Indicates the steepness of a line.
- Slope Formula: Calculated as ((y1 - y2) / (x1 - x2)).
- X-intercept: Point where the line crosses the x-axis ((x, 0)).
- Y-intercept: Point where the line crosses the y-axis ((0, y)).
Special Forms and Systems
- Slope-Intercept Form: (y = mx + b).
- Point-Slope Form: ((y - y1) = m(x - x1)).
- Absolute Value Function: Involves expressions containing absolute values.
- Scatter Plot: Graph shows potential relationships between two data sets.
Systems of Equations
- Solution of a System: Any ordered pair that satisfies all equations in the system.
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Types of Systems:
- One Solution: Exact value solution (e.g., (x = 7)).
- No Solutions: Contradictory statements (e.g., (1 = 4)).
- Many Solutions: Identities (e.g., (5 = 5)).
Exponential Functions
- Exponential Function: Defined by (f(x) = ab^x).
- Exponent: Indicates how many times the base is multiplied by itself.
Polynomials
- Monomial: A single term consisting of numbers and variables with whole number exponents.
- Binomial: A polynomial with two terms.
- Trinomial: A polynomial with three terms.
- Polynomial Definition: Composed of one or more terms.
Exponent Rules
- Multiply Powers: Applies when multiplying like bases, e.g., (x^2 \cdot x^2 = x^4).
- Power of Power: ((x^3)^2 = x^6).
- Power of Product: ((2ab)^2 = 4a^2b^2).
- Negative Exponent: Represents reciprocal, e.g., (2^{-1} = \frac{1}{2}).
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