§ 6.2 Adding and Subtracting Rational Expressions

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  • 6.2 Adding and Subtracting Rational Expressions
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  • Blitzer, Intermediate Algebra, 5e Slide #2 Section 6.2 Adding Rational Expressions In this section, you will practice adding and subtracting rational expressions. Remember that when adding or subtracting fractions, it is necessary to rewrite the fractions as fractions having the same denominator, which is called the common denominator for the fractions being combined.
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  • Blitzer, Intermediate Algebra, 5e Slide #3 Section 6.2 Adding Rational Expressions Adding Rational Expressions With Common Denominators If are rational expressions, then To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.
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  • Blitzer, Intermediate Algebra, 5e Slide #4 Section 6.2 Adding Rational ExpressionsEXAMPLE Add: SOLUTION Add numerators. Place this sum over the common denominator. Factor. This is the original expression. Combine like terms.
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  • Blitzer, Intermediate Algebra, 5e Slide #5 Section 6.2 Adding Rational Expressions Simplify. Factor and simplify by dividing out the common factor, x. CONTINUED
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  • Blitzer, Intermediate Algebra, 5e Slide #6 Section 6.2 Adding Rational Expressions Check Point 1 Add: SOLUTION Add numerators. Place this sum over the common denominator. Factor. Combine like terms. Simplify.
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  • Blitzer, Intermediate Algebra, 5e Slide #7 Section 6.2 Subtracting Rational Expressions Subtracting Rational Expressions With Common Denominators If are rational expressions, then To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, simplify the result.
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  • Blitzer, Intermediate Algebra, 5e Slide #8 Section 6.2 Subtracting Rational ExpressionsEXAMPLE Subtract: SOLUTION Subtract numerators. Place this difference over the common denominator. This is the original expression. Remove the parentheses and distribute.
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  • Blitzer, Intermediate Algebra, 5e Slide #9 Section 6.2 Subtracting Rational Expressions Factor. Combine like terms. Factor and simplify by dividing out the common factor, x + 3. CONTINUED Simplify.
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  • Blitzer, Intermediate Algebra, 5e Slide #10 Section 6.2 Adding Rational Expressions Check Point 2 Subtract: SOLUTION Subtract numerators. Place difference over the common denominator. Factor. Combine like terms. Simplify.
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  • Blitzer, Intermediate Algebra, 5e Slide #11 Section 6.2 Least Common Denominators Finding the Least Common Denominator (LCD) 1) Factor each denominator completely. 2) List the factors of the first denominator. 3) Add to the list in step 2 any factors of the second denominator that do not appear in the list. 4) Form the product of each different factor from the list in step 3. This product is the least common denominator.
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  • Blitzer, Intermediate Algebra, 5e Slide #12 Section 6.2 Least Common DenominatorsEXAMPLE Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator.
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  • Blitzer, Intermediate Algebra, 5e Slide #13 Section 6.2 Least Common Denominators 3) Add any unlisted factors from the second denominator. The second denominator is (2y - 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y 1 is not. We add the factor 2y 1 to our list. 4) The least common denominator is the product of all factors in the final list. Thus, CONTINUED is the least common denominator.
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  • Blitzer, Intermediate Algebra, 5e Slide #14 Section 6.2 Least Common Denominators Check Point 3 Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator. 3) Add any unlisted factors from the second denominator. 4) The least common denominator is the product of all factors in the final list.
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  • Blitzer, Intermediate Algebra, 5e Slide #15 Section 6.2 Least Common Denominators Check Point 4 Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator. 3) Add any unlisted factors from the second denominator. 4) The least common denominator is the product of all factors in the final list.
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  • Blitzer, Intermediate Algebra, 5e Slide #16 Section 6.2 Add & Subtract Fractions Adding and Subtracting Rational Expressions That Have Different Denominators 1) Find the LCD of the rational expressions. 2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD. 3) Add or subtract numerators, placing the resulting expression over the LCD. 4) If possible, simplify the resulting rational expressions.
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  • Blitzer, Intermediate Algebra, 5e Slide #17 Section 6.2 Adding FractionsEXAMPLE Add: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x + 4 and x 2. The only factor from the second denominator that is unlisted is x 1. Thus, the least common denominator is,
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  • Blitzer, Intermediate Algebra, 5e Slide #18 Section 6.2 Adding Fractions 2) Write equivalent expressions with the LCD as denominators. This is the original expression. CONTINUED Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.
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  • Blitzer, Intermediate Algebra, 5e Slide #19 Section 6.2 Adding Fractions 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Add numerators. CONTINUED Perform the multiplications using the distributive property. Combine like terms.
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  • Blitzer, Intermediate Algebra, 5e Slide #20 Section 6.2 Adding FractionsCONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
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  • Blitzer, Intermediate Algebra, 5e Slide #21 Section 6.2 Adding Fractions Check Point 5 Add SOLUTION Multiply Like fractions Simplified
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  • Blitzer, Intermediate Algebra, 5e Slide #22 Section 6.2 Adding Fractions Check Point 6 Add SOLUTION Multiply Like fractions Simplify Combine like terms
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  • Blitzer, Intermediate Algebra, 5e Slide #23 Section 6.2 Subtracting FractionsEXAMPLE Subtract: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x - 4 and x 1. The only factor from the second denominator that is unlisted is x + 1. Thus, the least common denominator is,
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  • Blitzer, Intermediate Algebra, 5e Slide #24 Section 6.2 Subtracting Fractions 2) Write equivalent expressions with the LCD as denominators. This is the original expression. CONTINUED Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.
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  • Blitzer, Intermediate Algebra, 5e Slide #25 Section 6.2 Subtracting Fractions 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Subtract numerators. CONTINUED Perform the multiplications using the distributive property and FOIL. Remove parentheses.
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  • Blitzer, Intermediate Algebra, 5e Slide #26 Section 6.2 Subtracting Fractions Combine like terms in the numerator. CONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
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  • Blitzer, Intermediate Algebra, 5e Slide #27 Section 6.2 Subtracting Fractions Check Point 7 Subtract SOLUTION Multiply Like fractions Combine like terms and Simplify Subtract (add the opposite)
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  • Blitzer, Intermediate Algebra, 5e Slide #28 Section 6.2 Adding and Subtracting of FractionsEXAMPLE Perform the indicated operations: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are 1 and x 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,
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  • Blitzer, Intermediate Algebra, 5e Slide #29 Section 6.2 Adding and Subtracting of FractionsCONTINUED 2) Write equivalent expressions with the LCD as denominator. This is the original expression. Factor the second denominator. Multiply each numerator and denominator by the extra factor required to form the LCD.
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  • Blitzer, Intermediate Algebra, 5e Slide #30 Section 6.2 Adding and Subtracting of FractionsCONTINUED 3) & 4) Add and subtract numerators, putting this result over the LCD. Simplify if possible. Add and subtract numerators. Perform the multiplications using the distributive property. Combine like terms in the numerator.
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  • Blitzer, Intermediate Algebra, 5e Slide #31 Section 6.2 Adding and