Basic Math and Pre-Algebra - Preliminaries - Quick Guides

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Study GuideBasic Math and Pre-Algebra–Preliminaries1. MulƟplying and Dividing Using ZeroIt zero works in multiplication and division. Zero may seem simple, but it follows some very importantrules!1.1MulƟplying by ZeroHere’s the key rule:Any number multiplied by zero equals zero.It doesn’t matter how big or small the number is. If one of the factors is 0, the product will always be 0.Examples:•( 0 × 2 = 0 )•( 8 × 2 × 3 × 6 × 0 = 0 )Even in a long multiplication problem, if just one number is 0, the entire answer becomes 0.Why?Because multiplying means combining equal groups. If you have zero groups of something, you havenothing. So the result is zero.1.2Dividing Zero by a NumberNow let’s talk about division.If you dividezero by any nonzero number, the answer iszero.Examples:•( 0 ÷ 3 = 0 )•(0/7 = 0 )

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Study GuideWhy?Division asks, “How many times does this number fit into another number?”If there’s nothing (zero) to divide, then the answer is zero.1.3Important: Dividing by ZeroDividingbyzero is not allowed.It is called“undefined.”Examples that arenot permitted:•(4/0)•(0/0)These do not have logical answers. The result isnot zero—it simply doesn’t exist. That’s why we saydividing by zero is undefined.Summary•Multiply any number by 0→Answer is 0•Divide 0 by any nonzero number→Answer is 0•Divide by 0→Undefined (not allowed)2. Common Math SymbolsIn math and algebra, symbols help us compare numbers and show relationships clearly. Learningwhat these symbols mean will make solving problems much easier. Let’s go through the mostcommon ones step by step.

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Study Guide2.1Equality SymbolsThese symbols show whether two values are the same or not.•=meansis equal to•Example: ( 5 + 3 = 8 )•≠meansis not equal to•Example: ( 4≠7 )2.2Greater Than and Less ThanThese symbols compare the size of numbers.•>meansis greater than•Example: ( 9 > 4 )•<meansis less than•Example: ( 2 < 6 )A helpful tip:The wide opening of the symbol always points toward the larger number.2.3Greater Than or Equal To / Less Than or Equal ToSometimes numbers can be equal, too. These symbols include that possibility.•≥meansis greater than or equal to•(Also written as≥)•Example: ( x≥5 )•≤meansis less than or equal to•(Also written as≤)•Example: ( y≤10 )

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Study Guide2.4“Not” Comparison SymbolsThese symbols show that something isnottrue.•≯meansis not greater than•≮meansis not less than•≱meansis not greater than or equal to•≰meansis not less than or equal toThese are often used in higher-level math to clearly state what is not true.2.5Approximate EqualitySometimes a value isn’t exact, but it’s close.•≈meansis approximately equal to•(Also written as≅in some cases)•Example: (π≈3.14 )This symbol is useful when rounding numbers or working with estimates.Summary•=equal to•≠not equal to•>greater than•<less than•≥greater than or equal to•≤less than or equal to•≈approximately equal to

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Study Guide3. Quiz: Ways to Show MulƟplicaƟon and Division, MulƟplying andDividing by Zero, and Common Math Symbols1. QuestionWhich of the following isNOTan acceptable way to show the multiplication of 5 with 4?Answer Choices•(5)(4)•5/4•5·4Correct Answer࿨࿩࿪5/4Why This Is CorrectMultiplication can be shown in several correct ways, such as:•Using parentheses: (5)(4)•Using a dot: 5 · 4However,5/4represents division, not multiplication.So it isnotan acceptable way to show 5 multiplied by 4.2. QuestionWhich of the following is an acceptable way to show the division of 12 with 5?Answer Choices•12(5)•(12)(5)•12/5

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Study GuideCorrect Answer࿨࿩࿪12/5Why This Is CorrectDivision is properly shown using the division symbol (÷) or a fraction bar (/).•12/5clearly shows 12 divided by 5.•12(5) and (12)(5) both represent multiplication, not division.3. Question0 ÷ 3 = ?Answer Choices•0•no such answer•3Correct Answer࿨࿩࿪0Why This Is CorrectZero divided by any nonzero number equals zero.Think of it this way:If you have 0 objects and divide them into 3 groups, each group still has 0 objects.4. Question3 ÷ 0 = ?

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Study GuideAnswer Choices•3•0•no such answerCorrect Answer࿨࿩࿪no such answerWhy This Is CorrectDivision by zero isundefined.There is no number you can multiply by 0 to get 3.Since it's impossible, the answer does not exist.5. QuestionWhich of the following is a true statement?Answer Choices•12 × 3 × 0 × 1 = (5 ÷ 0)•0 ÷ (12) = 0 × 12•0/0 = 0Correct Answer࿨࿩࿪0 ÷ (12) = 0 × 12Why This Is Correct•12 × 3 × 0 × 1 = 0, but 5 ÷ 0 is undefined→ᤶᤷNot equal•0 ÷ 12 = 0 and 0 × 12 = 0→࿨࿩࿪True•0/0 is undefined→ᤶᤷFalseOnly the second statement is true.

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Study Guide6. QuestionWhich symbol has exactly the same meaning as “is not greater than”?Answer Choices•<•>•≤Correct Answer࿨࿩࿪≤Why This Is Correct“Is not greater than” means the number is either:•Less than•Or equal toThe symbol≤means “less than or equal to,” which matches perfectly.7. QuestionWhich of the following statements is true?Answer Choices•12≰11•3≰5•6≱6

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Study GuideUnderstanding the Symbols•≤meansless than or equal to•≥meansgreater than or equal to•≰meansnot less than or equal to•≱meansnot greater than or equal toThe slash means the statement is saying the opposite.Check Each Statement12≰11This means:“12 isnotless than or equal to 11.”Since 12 is greater than 11, it is true that 12 isnotless than or equal to 11.࿨࿩࿪True3≤5This means:“3 is less than or equal to 5.”Since 3 is less than 5, this is also࿨࿩࿪True6≱6This means:“6 isnotgreater than or equal to 6.”But 6is equal to 6, so 6≥6 is true.That means saying it isnotgreater than or equal to 6 isᤶᤷFalse

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Study GuideFinal AnswerBoth of these statements are true:࿨࿩࿪12≰11࿨࿩࿪3≤5The only false statement is:ᤶᤷ6≱68. QuestionWhich statement says “x is approximately equal to 5”?Answer Choices•x≈5•x = 5•x≠5Correct Answer࿨࿩࿪x≈5Why This Is CorrectThe symbol≈means “approximately equal to.”•x = 5 means exactly equal•x≠5 means not equal•x≈5 means close to 5, but not exactly 5

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