Honors Algebra 1 Final Review
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Questions and Answers

What are natural numbers?

  • Any number that can be written as a quotient of 2 integers
  • All positive numbers including 0
  • All positive and negative whole numbers
  • All positive numbers not including 0 (correct)

Which of the following includes 0?

  • Natural numbers (N)
  • Rational numbers (Q)
  • Integers (Z)
  • Whole numbers (W) (correct)

What are integers?

  • All whole numbers including fractions
  • All positive numbers
  • All positive and negative whole numbers (correct)
  • Any number that can be expressed as a decimal

Which number type can be written as a quotient of two integers?

<p>Rational numbers (Q) (D)</p> Signup and view all the answers

What are irrational numbers?

<p>Any number with pi or the square root of a non-perfect number.</p> Signup and view all the answers

What is an imaginary number?

<p>Any equation where the negative is inside the square root sign.</p> Signup and view all the answers

What does 'element of' mean?

<p>Member of a set.</p> Signup and view all the answers

Union sets together combine two sets.

<p>True (A)</p> Signup and view all the answers

Intersection includes numbers not found in either of the combined sets.

<p>False (B)</p> Signup and view all the answers

What is a subset?

<p>Every number in this set is also a number in another set.</p> Signup and view all the answers

What makes up the real number system?

<p>Rational numbers, natural numbers, whole numbers, integers, and irrational numbers.</p> Signup and view all the answers

What does a null set imply?

<p>No numbers in this set.</p> Signup and view all the answers

What is the identity property for addition?

<p>Any number plus 0 equals that number (A)</p> Signup and view all the answers

What is the property of 0 in multiplication?

<p>Any number times 0 equals 0 (A)</p> Signup and view all the answers

What does substitution mean in mathematics?

<p>Any number can be replaced by another number.</p> Signup and view all the answers

What is the reflexive property?

<p>Both sides of the equation mirror each other.</p> Signup and view all the answers

What is the symmetric property?

<p>If a = b, then b = a (B)</p> Signup and view all the answers

What does transitive property mean?

<p>If a = b, and b = c, then a = c.</p> Signup and view all the answers

What is an inverse in multiplication?

<p>Any number times its reciprocal equals 1.</p> Signup and view all the answers

What does the definition of subtraction suggest?

<p>Subtraction can be changed to adding a negative (B)</p> Signup and view all the answers

What does the distributive property involve?

<p>Distributing everything outside parentheses to terms inside (B)</p> Signup and view all the answers

The requirements for simplest form are that _____

<p>all like terms are combined and all parentheses have been removed.</p> Signup and view all the answers

Which of the following phrases also means add?

<p>Plus, sum (D)</p> Signup and view all the answers

What is the phrase associated with subtraction?

<p>Minus, difference (B)</p> Signup and view all the answers

What words are associated with multiplication?

<p>Times, product (B)</p> Signup and view all the answers

What words are associated with division?

<p>Quotient, half (B)</p> Signup and view all the answers

Translate the verbal expression 'm increased by 5' into an algebraic expression.

<p>m + 5</p> Signup and view all the answers

Translate '7 less than a number' into an algebraic expression.

<p>x - 7</p> Signup and view all the answers

Translate 'the product of 5 and a' into an algebraic expression.

<p>5a</p> Signup and view all the answers

Translate 'the quotient of x and y' into an algebraic expression.

<p>x/y</p> Signup and view all the answers

Translate '6 times the sum of 10 and b' into an algebraic expression.

<p>6(10 + b)</p> Signup and view all the answers

Translate '(3 + b) divided by y' into an algebraic expression.

<p>(3 + b)/y</p> Signup and view all the answers

Translate '9 times t' into an algebraic expression.

<p>9t</p> Signup and view all the answers

Translate '5 times y plus 10 times x' into an algebraic expression.

<p>5y + 10x</p> Signup and view all the answers

Translate 'the quantity of 15 and n times 2' into an algebraic expression.

<p>2(15 + n)</p> Signup and view all the answers

Translate 's subtracted from 8 times r' into an algebraic expression.

<p>8r - s</p> Signup and view all the answers

Positive times a negative equals _____

<p>a negative.</p> Signup and view all the answers

Positive times a positive equals _____

<p>a positive.</p> Signup and view all the answers

Negative times a negative equals _____

<p>a positive.</p> Signup and view all the answers

What does the term 'domain' refer to?

<p>Another word for x values.</p> Signup and view all the answers

What does the term 'range' refer to?

<p>Another word for y values.</p> Signup and view all the answers

What is a relation in mathematics?

<p>A set of ordered pairs.</p> Signup and view all the answers

What are four ways to represent relations?

<p>Order pairs, table, mapping, graphs (B)</p> Signup and view all the answers

A function assigns exactly one output value for each input value.

<p>True (A)</p> Signup and view all the answers

What is an independent variable?

<p>The thing being changed.</p> Signup and view all the answers

What is a dependent variable?

<p>The thing being measured.</p> Signup and view all the answers

What is a function rule?

<p>Equation that describes the function relationship.</p> Signup and view all the answers

What does the addition property state?

<p>If a = b, then a + c = b + c (A)</p> Signup and view all the answers

What does the multiplication property state?

<p>If a = b, then a c = b c (D)</p> Signup and view all the answers

What does the subtraction property state?

<p>If a = b, then a - c = b - c (B)</p> Signup and view all the answers

What does the division property state?

<p>If a = b, then a / c = b / c, c cannot equal 0 (A)</p> Signup and view all the answers

Properties of equality are used _____ when isolating the variable.

<p>to maintain balance in an equation.</p> Signup and view all the answers

When solving equations with variables on both sides, _____

<p>move the variable first.</p> Signup and view all the answers

What signifies infinitely many solutions in equations?

<p>Variables fall out and leave you with a true statement (like x = x).</p> Signup and view all the answers

What indicates no solution in equations?

<p>Variables fall out and leave you with a false statement (like x = y).</p> Signup and view all the answers

What is a ratio?

<p>A comparison of two numbers by division.</p> Signup and view all the answers

Translate '2:3' into simplest form for the ratio '32:48'.

<p>1:1.5</p> Signup and view all the answers

Translate '1:3' into simplest form for the ratio '5:15'.

<p>1:3</p> Signup and view all the answers

What is a proportion?

<p>An equation where two ratios are equal.</p> Signup and view all the answers

What should you do when solving a proportion?

<p>Cross multiply.</p> Signup and view all the answers

How do you calculate the final price after a discount?

<p>Multiply the percentage decimal by the original price and subtract that from the original price.</p> Signup and view all the answers

How do you calculate the final price after tax?

<p>Multiply the percentage decimal by the original price and add that to the original price.</p> Signup and view all the answers

What is the equation used to find percentage of change?

<p>percent of change/100 = new - original/original.</p> Signup and view all the answers

What are variables for finding consecutive integers?

<p>x, x + 1, x + 2, etc.</p> Signup and view all the answers

What are variables for finding consecutive even or odd integers?

<p>x, x + 2, x + 4, etc.</p> Signup and view all the answers

What are the headings for mixture problems?

<p>What, cost per item, quantity, total value (B)</p> Signup and view all the answers

What are the headings for mixture problems involving percentages?

<p>What, percentage, quantity, total value (C)</p> Signup and view all the answers

What are the headings for mixture problems involving interest?

<p>What, amount invested, rate, interest (C)</p> Signup and view all the answers

What equation governs motion problems?

<p>Rate × Time = Distance.</p> Signup and view all the answers

What are three types of motion word problems?

<p>Opposite direction, round trip, leaving later (A)</p> Signup and view all the answers

What are the headers for work word problems?

<p>Who, rate, time, work (D)</p> Signup and view all the answers

Is fewer than means _____

<p>less than.</p> Signup and view all the answers

Is more than means _____

<p>greater than.</p> Signup and view all the answers

What phrases mean less than or equal to?

<p>At most, no more than (B)</p> Signup and view all the answers

What phrases mean greater than or equal to?

<p>At least, no greater than (B)</p> Signup and view all the answers

What should you do when solving a proportion with an inequality symbol?

<p>You don't cross multiply; you clear the fractions.</p> Signup and view all the answers

What does {x|x>y} signify?

<p>Example of set notation.</p> Signup and view all the answers

What does a compound inequality involve?

<p>Two inequalities connected by and/or.</p> Signup and view all the answers

What does disjunction represent?

<p>All of them together (D)</p> Signup and view all the answers

What does conjunction represent?

<p>Intersection with 'and' (A)</p> Signup and view all the answers

When multiplying or dividing by a negative number, you must _____

<p>flip the inequality symbol.</p> Signup and view all the answers

The absolute value can be equal to a negative number.

<p>False (B)</p> Signup and view all the answers

What kind of compound inequality is associated with the less than symbol?

<p>Conjunction (C)</p> Signup and view all the answers

What kind of compound inequality is associated with the greater than symbol?

<p>Disjunction (D)</p> Signup and view all the answers

What is an extraneous solution?

<p>Solution to the equation that does not satisfy the original equation.</p> Signup and view all the answers

What do you do when performing unit conversions?

<p>Eliminate all labels other than the ones you are using in your final answer through cross simplification.</p> Signup and view all the answers

What does f(x) represent?

<p>Example of function notation.</p> Signup and view all the answers

What does the y value represent in functions?

<p>What does f(x) really mean?</p> Signup and view all the answers

What is a discrete function?

<p>Function where numbers in between points don't work (D)</p> Signup and view all the answers

What is a continuous function?

<p>Function where everything in between the two points works (A)</p> Signup and view all the answers

What is the median?

<p>Middle number.</p> Signup and view all the answers

What is the mean?

<p>Average.</p> Signup and view all the answers

What is the mode?

<p>Most common number.</p> Signup and view all the answers

How do you find the median?

<p>Line your numbers up in numerical order and find the middle number.</p> Signup and view all the answers

Flashcards

Natural Numbers (N)

All positive whole numbers, excluding zero. (e.g., 1, 2, 3, ...)

Whole Numbers (W)

All positive whole numbers, including zero. (e.g., 0, 1, 2, 3, ...)

Integers (Z)

All positive and negative whole numbers, including zero. (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...)

Rational Numbers (Q)

Numbers that can be expressed as a fraction (a/b), where a and b are integers, and b is not zero.

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Irrational Numbers (I)

Numbers that cannot be expressed as fractions, involving roots of non-perfect squares or π.

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Imaginary Numbers

Numbers resulting from the square root of a negative number, denoted by 'i' where i² = -1.

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Element Of

Describes a member that belongs to a set. Represented by the symbol '∈'.

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Union

Combines two sets, including all elements from both sets. Represented by the symbol '∪'.

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Intersection

Includes only the elements common to both sets. Represented by the symbol '∩'.

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Subset

Every element in the subset is also in another set. Represented by the symbol '⊆'.

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Null Set

A set with no elements. Represented by '{}' or '∅'.

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Identity (+)

Adding 0 to any number leaves it unchanged. (e.g., a + 0 = a)

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Identity (×)

Multiplying any number by 1 leaves it unchanged. (e.g., a × 1 = a)

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Property of 0

Multiplying any number by 0 results in 0. (e.g., a × 0 = 0)

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Substitution Property

A number can be replaced by another in an equation if they are equal. (e.g., if a = b, then a can be replaced by b)

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Reflexive Property

An equation is equal to itself. (e.g., a = a)

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Symmetric Property

If a = b, then b = a.

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Transitive Property

If a = b and b = c, then a = c.

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Inverse (+)

For any number 'a', there exists an additive inverse '(-a)' such that a + (-a) = 0.

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Inverse (×)

For any non-zero number 'a', there exists a multiplicative inverse '1/a' such that a × (1/a) = 1.

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Definition of Subtraction

Subtracting a number is the same as adding its additive inverse.

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Distributive Property

a(b + c) = ab + ac

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Commutative Property

The order of numbers in addition or multiplication does not affect the result.

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Associative Property

Grouping of numbers in addition or multiplication does not change the result.

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Consecutive Integers

Integers that follow each other in order, represented as x, x+1, x+2, ...

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Consecutive Odd/Even Integers

Odd/Even numbers that follow each other in order, represented as x, x+2, x+4, ...

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Inequality

A mathematical statement that compares two values using symbols like <, >, ≤, ≥.

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Proportion

An equation stating that two ratios are equal. (e.g., a/b = c/d)

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Cross Multiplication

A method used to solve proportions by multiplying the numerator of one fraction by the denominator of the other.

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Function

A relation assigning exactly one output for each input. Represented by f(x).

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Independent Variable

The variable that is changed or controlled in an experiment.

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Dependent Variable

The variable that is measured or affected by the independent variable.

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Mean

The average of a set of numbers. Calculated by summing all the numbers and dividing by the number of elements.

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Median

The middle value in a sorted list of numbers. If there are two middle values, find their average.

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Mode

The most frequently occurring number in a dataset.

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Study Notes

Number Sets

  • Natural Numbers (N): All positive numbers, excluding 0 (e.g., 1, 2, 3).
  • Whole Numbers (W): All positive numbers, including 0 (e.g., 0, 1, 2).
  • Integers (Z): All positive and negative whole numbers (e.g., -1, 0, 1).
  • Rational Numbers (Q): Numbers that can be expressed as a quotient of two integers (e.g., 1/2, 2.4, 5).
  • Irrational Numbers (I): Numbers that cannot be expressed as fractions, involving roots of non-perfect squares or π.
  • Imaginary Numbers: Numbers resulting from the square root of a negative number.

Set Operations

  • Element Of: Describes a member of a set.
  • Union: Combines two sets, including all elements from both.
  • Intersection: Includes only the elements common to both sets.
  • Subset: Every element in the subset is also in another set.
  • Null Set: A set with no elements.

Algebraic Properties

  • Identity (+): Adding 0 to any number leaves it unchanged.
  • Identity (×): Multiplying any number by 1 leaves it unchanged.
  • Property of 0: Multiplying any number by 0 results in 0.
  • Substitution: A number can be replaced by another in an equation.
  • Reflexive Property: An equation is equal to itself.
  • Symmetric Property: If a = b, then b = a.
  • Transitive Property: If a = b and b = c, then a = c.
  • Inverse (+): A number plus its additive inverse equals 0.
  • Inverse (×): A number times its reciprocal equals 1.

Operations and Properties

  • Definition of Subtraction: Subtraction can be viewed as adding a negative.
  • Distributive Property: Distributing terms in parentheses to each other.
  • Commutative Property: The order of numbers does not affect the sum or product.
  • Associative Property: Grouping of numbers in addition or multiplication does not change the result.

Algebraic Expressions

  • Understanding how to translate verbal expressions into algebraic forms and vice versa.
  • Consecutive Integers: Identified using expressions like x, x+1, etc.
  • Consecutive Odd/Even Integers: Expressed as x, x+2, x+4, etc.

Inequalities and Proportions

  • Inequalities can be expressed in various ways, such as "is less than" or "is more than."
  • Proportion: An equation that states that two ratios are equal.
  • Cross Multiplication: A method used to solve proportions.

Function Basics

  • Function: A relation assigning exactly one output for each input.
  • Independent Variable: The variable that is changed or controlled in an experiment.
  • Dependent Variable: The variable that is measured or affected.
  • Function Notation: Typically written as f(x), indicating the output value when x is substituted.

Statistical Concepts

  • Mean: The average of a set of numbers.
  • Median: The middle value in a sorted list of numbers.
  • Mode: The most frequently occurring number in a dataset.

Special Concepts

  • Extraneous Solution: A solution derived from the equation that does not satisfy the original context.
  • Continuous Function: A function where every value between two points is included.
  • Discrete Function: A function where only specific points are defined and values in between do not apply.

Problem-Solving Techniques

  • Employ properties of equality for solving equations with variables.
  • The method of unit conversions requires careful attention to labels to maintain consistency in answers.

These notes combine definitions, properties, and explanations of fundamental concepts in algebra to aid in review and understanding.

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Prepare for your Honors Algebra 1 final exam with this comprehensive review of key concepts from the first semester. This quiz covers essential definitions such as natural numbers, whole numbers, integers, and rational numbers, providing you with a solid foundation in number types. Test your knowledge and ensure you're ready for the exam!

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