Section 7.1 Rational Functions and Simplifying Rational Expressions
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1 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Complete the outline as you view Video Lecture 7.1. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Find the domain of a rational function A rational expression can be written as P Q, where P and Q are polynomials, Q 0. A rational function is of the form f( x) P Q as long as P Q is a rational expression. Find the domain of the rational function. 1. 3x f( x) 7 x Pause and work. x 3. Cx ( ) x 4 Play and check. 159
2 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Objective Simplify or write rational expressions in lowest terms To Simplify a Rational Expression Step 1: Completely the numerator and denominator. Step : factors common to the numerator and denominator. (This is the same as removing a factor of 1. ) Simplify. 3. 5a5b a b Simplify. Pause and work. 4. x 7 7 x Does order matter when adding? (Addition is commutative.) Play and check. 160
3 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Simplify. 5. x 7 7 x Does order matter when subtracting? (Subtracting is not commutative.) a b 1 (denominator not 0) b a Simplify x 7x x 5x14 Pause and work x x Play and check. 161
4 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Objective 3 Write equivalent forms of rational expressions Write equivalent rational expressions for the given rational expression. 8. x 11 x 4 Objective 4 Use rational functions in applications 9. The total revenue from the sale of a popular book is approximated by the rational function 1000x Rx ( ) x 4, where x is the number of years since publication and R(x) is the total revenue in millions of dollars. a. Find the total revenue at the end of the first year. b. Find the total revenue at the end of the second year. c. Find the revenue during the second year only. d. Find the domain of the function. 16
5 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7. Section 7. Multiplying and Dividing Rational Expressions Complete the outline as you view Video Lecture 7.. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Multiply rational expressions Multiply x x 4x Multiplying Rational Expressions If P Q and R S P R PR are rational expressions, then Q S QS. Multiplying Rational Expressions Step 1: Completely numerators and denominators. Step : Multiply and multiply. Step 3: or write the product in lowest terms by dividing out common factors. Multiply.. 5x0 3x 13x4 3x x x
6 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7. Section 7. Multiplying and Dividing Rational Expressions Objective Divide rational expressions Dividing Rational Expressions If P Q and R S are rational expressions, and R S is not 0, then P R P S PS Q S Q R QR To divide by a rational expression by its. Divide. 3. x 5 6 x x 7x x 9x14 Objective 3 Multiply or divide rational expressions Divide. Pause and work x 10 4 x Play and check. 164
7 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7. Find the area of the rectangle. Section 7. Multiplying and Dividing Rational Expressions 5. x x 5 ft A lw x 5 9x ft Objective 4 Convert between units of measure Convert the units of measure cubic yards = cubic feet 1 cubic yard 165
8 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7. Section 7. Multiplying and Dividing Rational Expressions 166
9 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.3 Section 7.3 Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator Complete the outline as you view Video Lecture 7.3. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Add and subtract rational expressions with the same denominator Add the rational expressions x7 x7 Adding and Subtracting Rational Expressions with Common Denominators If P R and Q are rational expressions, then R P Q P Q and R R R P Q P Q R R R. Add.. Pause and work. 9 y 1 3 y 3 y Play and check. Can divide out common. Cannot divide out common. 167
10 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.3 Section 7.3 Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator To simplify: Step 1: Factor and. Step : common factors. Subtract. x3 x x x30 x x30 3. Objective Find the least common denominator of a list of rational expressions Finding the Least Common Denominator (LCD) Step 1: Factor each completely. Step : The least common denominator (LCD) is the of all unique factors found in Step 1, each raised to a power equal to the greatest number of times that the factor appears in any one factored denominator. Find the least common denominator (LCD) x, 3 x 4 8x x 4 LCD 168
11 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.3 Section 7.3 Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator 5. Pause and work. 1 8, 3x 3 x 4x 3x 3 x 4x LCD Play and check. Objective 3 given Write a rational expression as an equivalent expression whose denominator is Write an equivalent rational expression with the given denominator a 1ab 7. 9 a 5a10 5 b( a) Equivalent rational expressions simplify to the same rational expression. 169
12 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.3 Section 7.3 Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator 170
13 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.4 Section 7.4 Adding and Subtracting Rational Expressions with Unlike Denominators Complete the outline as you view Video Lecture 7.4. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Add and subtract rational expressions with unlike denominators Add x x To add or subtract rational expressions, we must have a. Add x 3 3x Remember that a a a b b b. Subtract. 3. y y 3 171
14 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.4 Section 7.4 Adding and Subtracting Rational Expressions with Unlike Denominators Adding or Subtracting Rational Expressions with Unlike Denominators Step 1: Find the of the rational expressions. Step : Rewrite each rational expression as an expression whose denominator is the LCD found in Step 1. Step 3: Add or subtract and write the sum of difference over the common denominator. Step 4: or write the rational expression in simplest form. Subtract. 4. 3a a1 a6 a3 Add. Pause and work. x8 x1 x 5x6 x 4x5 5. Play and check. 17
15 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.5 Section 7.5 Solving Equations Containing Rational Expressions Complete the outline as you view Video Lecture 7.5. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Solve equations containing rational expressions Multiplication Property Multiply both sides by the same non-zero number. Solve. Check your solution. 1. x3 x 1 5 Pause and work y y Play and check. Be careful when there are variables in the denominator. Make sure the proposed solution does not make any denominator 0. If so, it is an extraneous solution. 173
16 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.5 Section 7.5 Solving Equations Containing Rational Expressions Solving an Equation Containing Rational Expressions Step 1: Multiply sides of the equation by the LCD of all rational expressions in the equation. Step : Remove any grouping symbols and solve the resulting equation. Step 3: the solution. Solve. Check your solution(s). 3. t t t Pause and work. 3 a a3 a3 Play and check. 4r r 5r14 r7 r 174
17 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.5 Section 7.5 Solving Equations Containing Rational Expressions Objective Solve equations containing rational expressions for a specified variable Solve for the indicated variable. u 6. T B E for B 175
18 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.5 Section 7.5 Solving Equations Containing Rational Expressions 176
19 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.6 Section 7.6 Proportion and Problem Solving with Rational Equations Complete the outline as you view Video Lecture 7.6. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Solve proportions A is the quotient of two quantities. A is a statement that two ratios are equal. Cross Products If a c b d, then ad bc. Use cross products to solve for x. 1. x Pause and work. x 1 x 3 3 Play and check. A proposed solution that makes the denominator 0 is called an solution. 177
20 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.6 Section 7.6 Proportion and Problem Solving with Rational Equations Objective Use proportions to solve problems Solve. 3. There are 110 calories per grams of Frosted Flakes cereal. Find out how many calories are in 1.5 grams of this cereal. Round to the nearest whole calorie. Find the unknown length x in the following pair of similar triangles. 4. Pause and work. Play and check. Objective 3 Solve problems about numbers 5. Twelve divided by the sum of x and equals the quotient of 4 and the difference of x and. Find x. 178
21 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.6 Section 7.6 Proportion and Problem Solving with Rational Equations Objective 4 Solve problems about work 6. In minutes, a conveyor belt moves 300 pounds of recyclable aluminum from the delivery truck to a storage area. A smaller belt moves the same quantity of cans the same distance in 6 minutes. If both belts are used, find how long it takes to move the cans to the storage area. belt smaller belt together minutes to complete job part of job completed in 1 minute Objective 5 Solve problems about distance 7. A car travels 80 miles in the same time that a motorcycle travels 40 miles. If the car s speed is 10 miles per hour more than the motorcycle s, find the speed of the car and the speed of the motorcycle. d = r t car motorcycle 179
22 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.6 Section 7.6 Proportion and Problem Solving with Rational Equations 180
23 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.7 Section 7.7 Simplifying Complex Fractions Complete the outline as you view Video Lecture 7.7. Pause the video as needed to fill in the blanks. Then press Play to continue. Also, circle your answer to each numbered exercise. Objective 1 Simplify complex fractions by simplifying the numerator and denominator and then dividing A fraction whose numerator or denominator or both contain rational expressions is a. Simplify the complex fraction x 5 6x Simplifying a Complex Fraction: Method 1 Step 1: Simplify the and the of the complex fraction so that each is a single fraction. Step : Perform the indicated division by multiplying the numerator of the complex fraction by the of the denominator of the complex fraction. Step 3: if possible. 181
24 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.7 Section 7.7 Simplifying Complex Fractions Pause and work.. Simplify the complex fraction 4x y xy 1 y x. Play and check. Objective Simplify complex fractions by multiplying by a common denominator Simplifying a Complex Fraction: Method Step 1: Multiply the numerator and denominator of the complex fraction by the of the fractions in both the and. Step : Simplify. 3. Simplify the complex fraction x x x 1. x1 x1 x x 1 18
25 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.7 Section 7.7 Simplifying Complex Fractions Objective 3 Simplify expressions with negative exponents Simplify. Write the expression with positive exponents first. 4. a 3b a b
26 Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.7 Section 7.7 Simplifying Complex Fractions 184
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