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Algebra I – Algebraic Fractions - Document preview page 1

Algebra I – Algebraic Fractions - Page 1

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Algebra I – Algebraic Fractions

This document provides study materials related to Algebra I – Algebraic Fractions. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

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Algebra I – Algebraic Fractions - Page 1 preview imageStudy GuideAlgebra IAlgebraic FracƟons1. What Are Algebraic FracƟons?Algebraic fractions are just like normal fractionsbut they includevariables(letters like x, a, or b).For example:Here,xis a variable in the denominator.So, an algebraic fraction is any fraction that has:A variable in the numerator, orA variable in the denominator, orVariables in both.Why Do We Have Restrictions?You already know something very important:Division by 0 is impossible.Because of this, any value that makes the denominator equal to0is not allowed.That means:The denominator of a fraction can never be zero.So whenever we see an algebraic fraction, we must check what values would make the denominatorzeroand exclude them.These excluded values are calledrestrictions.
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Algebra I – Algebraic Fractions - Page 2 preview imageStudy GuideLet’s Look at Some Examples1. Example:5/xSince x is in the denominator:Why?Because if x = 0, the fraction becomes:And that is undefined.So the restriction is:x cannot equal 02. Example:Here the denominator isx3.To find the restriction, ask:When does the denominator equal 0?
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Algebra I – Algebraic Fractions - Page 3 preview imageStudy GuideSo,x cannot equal 3We write:3. Example:Now the denominator isab.Set it equal to 0:[a-b = 0][a = b]So,a cannot equal bWe write:4.Example:The denominator is:[a2b]This expression equals 0 if:
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Algebra I – Algebraic Fractions - Page 4 preview imageStudy Guide(a = 0), or(b = 0)So neither value is allowed.a0 and b0Both variables must be nonzero.2. OperaƟons with Algebraic FracƟonsIntroductionWhen working withalgebraic fractions, the steps are very similar to working with regular fractions.The difference is that now we are working with variables (likex) along with numbers.In this chapter, you’ll learn how to:Reduce algebraic fractionsMultiply algebraic fractionsAvoid common mistakesAs you go through the examples, pay close attention to the steps. Learning the correct method willsave you time and help you avoid errors.Reducing Algebraic FractionsReducing means simplifying a fraction to its lowest terms.Step-by-Step Method:1.Factor the numerator(top part).2.Factor the denominator(bottom part).
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Algebra I – Algebraic Fractions - Page 5 preview imageStudy Guide3.Cancel common factors(only factorsnot individual terms).4.Write the simplified result.Very Important:You can only cancelfactors, not terms that are being added or subtracted.Example 11) Simplify:Step 1:Reduce the numbers4 ÷ 8 = 1/2Step 2:Reduce the variablesFinal Answer:2) Simplify:Step 1:Factor both numerator and denominator
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Algebra I – Algebraic Fractions - Page 6 preview imageStudy GuideStep 2:Cancel the common factor ((x-1))Final Answer:Notice we canceled theentire factor (x1)not just part of it.3) Simplify:Step 1:Factor both partsNumerator:Denominator:Step 2:Cancel the common factor ((x + 1))Final Answer:Warning: A Common Mistake
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Algebra I – Algebraic Fractions - Page 7 preview imageStudy GuideYoucannot reduce across addition or subtraction.For example:You cannot cancel the x’s.Also:You cannot cancel the 6 with part of the numerator.Always factor first.If something is being added or subtracted, donotcancel unless it is part of a common factor.Multiplying Algebraic FractionsMultiplying algebraic fractions is actually easier than it looks!Follow these steps:1.Factor all numerators.2.Factor all denominators.3.Cancel common factors.4.Multiply what remains on top.5.Multiply what remains on bottom.If you reduce correctly before multiplying, your final answer will already be simplified.Example 2:Multiplying Algebraic Fractions
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Algebra I – Algebraic Fractions - Page 8 preview imageStudy GuideHow to Multiply Algebraic FractionsMultiplying algebraic fractions is easier than it looks! Just follow these simple steps:Step-by-Step Method:1.Factor everything first(numerators and denominators).2.Cancel common factors.3.Multiply what remains on top.4.Multiply what remains on bottom.5.Simplify your final answer if needed.1) Multiply:Step 1:Multiply numeratorsStep 2:Multiply denominatorsFinal Answer:Simpleno common factors to cancel!
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Algebra I – Algebraic Fractions - Page 9 preview imageStudy Guide2) Multiply:Step 1:Multiply acrossStep 2:Cancel common factors(y) cancels with (y)One (x) cancels from (x2)Final Answer:Always cancel before multiplying completelyit keeps things easier!3) Multiply:Step 1:Factor everything
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Algebra I – Algebraic Fractions - Page 10 preview imageStudy GuideStep 2:Rewrite with factorsStep 3:Cancel common factorsCancel:((y+2))one ((x+1))Final Answer:4) Multiply:Step 1:Factor expressionsStep 2:Rewrite
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