Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10. The student will solve rational equations. (proportions and percents) Ratio: the relationship a b of two quantities, a and b, that are measured in the same units Proportion: two ratios that are equal In the proportion a b c, a and d are called the etremes; b and c are called the means d The product of the etremes is equal to the product of the means. ad bc When solving a proportion, cross multiply the two ratios. E: Solve the proportion 0 15. 15 5 0 0 4 E: Solve the equation 9 4. 49 6 ± 6 6, 6 Etraneous Solution: a value found when solving an equation that does not work in the original equation 1 E: Solve the equation + 4 + 4. Cross multiply. ( + ) ( + ) 4 1 4 + + 1 4 Solve. Check. 0 + 1 ( )( ) 0 + 4 1 4 + 4 ( 4) + 4( 4) 1 division by zero! 0 0 4, 1 + 4 + 4 1 true 1 7 Solution: (Note: 4 is an etraneous solution.) Application Problem E: An architect is creating a scale drawing of a room in a house. The room is 16 ft by 0.5 ft. The drawing will have a width of 8 inches. How long will the drawing be? Page 1 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Set up a proportion: width of room width of drawing length of room length of drawing 16 8 0.5 Solve the proportion: 16 164 10.5 in Percent Problems Percent Proportion: % part 100 whole E: 8 is 16% of what number? Set up a proportion: 16 8 Solve: 100 16 800 50 E: The price of a meal is $. Calculate a 15% tip for this meal. Set up a proportion: 15 Solve: 100 100 495 4.95 E: 6 is what percent of 0? Set up a proportion: 6 0 600 Solve: 100 0 10% You Try: Solve the proportion. 9 + QOD: Eplain the relationship of the means and etremes in a proportion. Page of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10.1 The student will simplify rational epressions with like and unlike denominators. Recall: When simplifying fractions, we divide out any common factors in the numerator and denominator E: Simplify 16 0. 4 4 The numerator and denominator have a common factor of 4. They can be rewritten. 45 Now we can divide out the common factor of 4. The remaining numerator and denominator have 4 4 4 no common factors (other than 1), so the fraction is now simplified. 4 5 5 Simplified Form of a Rational Epression: a rational epression in which the numerator and denominator have no common factors other than 1 Simplifying a Rational Epression 1. Factor the numerator and denominator. Divide out any common factors E: Simplify the epression Factor. Divide out common factors. 5 6. 1 ( 6)( + 1) ( + 1)( 1) ( 6) ( + 1) 6 ( + 1) ( 1 ) 1 Recall: When multiplying fractions, simplify any common factors in the numerators and denominators, then multiply the numerators and multiply the denominators. E: Multiply 5 4. Divide out common factors. 6 50 5 4 5 7 7 50 7 5 Multiplying Rational Epressions 1. Factor numerators and denominators (if necessary).. Divide out common factors.. Multiply numerators and denominators. Page of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations E: Multiply Factor. 7 4+ 1 1. 1 9 1 1 1 1+ 1 1 + 1 1 + 1 1 Divide out common factors. ( 1 ) ( 1+ ) ( + 1) ( 1) ( 1) ( 1) 1 1 1 1 1 Multiply. Dividing Rational Epressions Multiply the first epression by the reciprocal of the second epression and simplify. E: Divide. + 4 8 6 + Multiply by the reciprocal. + 6 4 8 + Factor and simplify. 4 ( ) ( + ) ( ) ( + ) 4 E: Find the quotient of 8 10 + and 4 4. Multiply by the reciprocal. 8 10 1 + 4 4 Factor and simplify. ( 4 1) ( + ) 1 ( ) 4 4 1 + 4 You Try: Simplify. ( ) 4 9 + QOD: When is a rational epression completely simplified? Page 4 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10.1 The student will simplify rational epressions with like and unlike denominators. (adding and subtracting) Recall: To add or subtract fractions with like denominators, add or subtract the numerators and keep the common denominator. Simplify the sum or difference. E: Find the difference. 8 15 15 8 6 15 15 5 Adding and Subtracting Rational Epressions with Like Denominators Add or subtract the numerators. Keep the common denominator. Simplify the sum or difference. E: Subtract. 7 7 4 E: Add. 6 + + + + ( + ) + ( + ) 6 Recall: To add or subtract fractions with unlike denominators, find the least common denominator (LCD) and rewrite each fraction with the common denominators. Then add or subtract the numerators and keep the common denominator. Simplify the sum or difference. E: Add. + 15 0 Note: To find the LCD, it is helpful to write the denominators in factored form. 15 5, 0 5 LCD 5 60 Adding and Subtracting Rational Epressions with Unlike Denominators 1. Find the least common denominator (in factored form).. Rewrite each fraction with the common denominator.. Add or subtract the numerators, keep the common denominator, and simplify. 4 E: Find the sum. + 6 + Factor and find the LCD. 4 + ( + ) 1 LCD ( + 1) Page 5 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Rewrite each fraction with the LCD. ( + 1) + ( + ) ( + ) ( + ) ( + ) 4 8 4 + + 1 1 1 1 Add the fractions. + 8+ 4 ( + ) 1 Note: Our answer cannot be simplified because the numerator cannot be factored. We will leave the denominator in factored form. + 1 1 E: Subtract: + 6+ 9 9 Factor and find the LCD. + 1 1 ( + ) ( + )( ) LCD ( + ) ( ) Rewrite each fraction with the LCD. + 1 1 + + + + + + + Subtract the fractions. 6 ( + ) ( ) ( + ) ( ) You Try: Perform the indicated operations and simplify. 4 5 + + 1 1 QOD: Eplain if the following is a true statement. The LCD of two rational epressions is the product of the denominators. Page 6 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 10. The student will solve rational equations. Recall: When solving equations involving fractions, we can eliminate the fractions by multiplying every term in the equation by the LCD. E: Solve the equation. 4 9 Multiply every term by the LCD 9. Solve the equation. 9 9 4 9 9 6 6 0 + 0 0 + 0 0 0, 0 Rational Equation: an equation that involves rational epression To solve a rational equation, multiply every term by the LCD. Then check your solution(s) in the original equation. E: Solve the equation 1 1 Multiply every term by the LCD Solve the equation. 6 4 18 18 1 1 Check. 1 1 1 1 true 18 18 6, so 18 E: Solve the equation 5 5 4 + 1 + 1 Multiply every term by the LCD + 1 ( +1) 5 ( + 1) 5 4+ 4 5 ( 1) 4 ( 1) + + 5 ( + 1) Solve the equation. 1 Page 7 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Check. 5( 1) 5 4 1 + 1 1 + 1 This solution leads to division by zero in the original equation. Therefore, it is an etraneous solution. This equation has no solution. 6 E: Solve the equation 1 + 4 Multiply every term by the LCD ( )( + ) ( ) ( + ) ( ) ( )( ) + + 4 4 6+ 4 6 ( )( + ) ( )( ) + + 1 Solve the equation. + 4 6 0 + 0 ( )( ) + 1 0, 1 Check. 6 + 1 4 11 6 + 1true 5 5 () () 1 6 + 1 1 1 4 1 6 + 1true 1 Solutions:, 1 8 5 You Try: Solve the equation: + 1 + 4 4 QOD: When is a solution an etraneous solution? Page 8 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Syllabus Objective: 11.1 The student will apply permutations and combinations to mathematical and practical situations, including the fundamental counting principle. Fundamental Counting Principle: the number of ways of multiple events is the product of the number of ways each event can occur E: A math student is going to flip a coin and roll a die. How many results are there for the coin flip and die roll? Number of results of the coin flip: (heads or tails) Number of results of the die roll: 6 (the numbers 1 6) Results of Both: 6 1 E: A meal deal at a restaurant consists of an appetizer, main course, and dessert. If the deal has appetizers, 5 main courses, and 6 desserts to choose from, how many different meal deals can be chosen? By the Fundamental Counting Principle, we will find the product of the number of each choices. 5 6 90 So there are 90 different meal deals. Permutation: the number of ordered arrangements of elements in a group E: A code for opening a safe consists of letters and digits. How many different possible codes are there for the safe? Write the number of possibilities for each part of the code: 6 6 10 10 10 letter letter digit digit digit Find the product: 6 6 10 10 10 676,000 There are 676,000 different safe codes. E: In a race with 5 runners, how many ways can the runners come in first, second, and third place? Write the number of possibilities for each place: 5 4 First Second Third Find the product: 5 4 1,800 There are 1,800 ways the runners can finish in. E: How many ways can the letters in the word UTAH be arranged? There are 4 letters to choose for the first position, for the second, for the third, and 1 for the fourth. Find the product: 4 1 4 Page 9 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8
Algebra I Notes Unit Thirteen: Rational Epressions and Equations Factorial: n n( n )( n )! 1... 1 E: 6! 6 54 1 70 Note how factorials would be used in the last eample of permutations. Think of a permutation problem whose solution would be 6!. E: How many ways can the letters in the word NEVADA be arranged? There are 6 letters in the word NEVADA, but are the same letter. So we must divide by to avoid counting the same word twice. 6541 60 Combination: the number of ways of picking elements for an unordered group E: How many groups of 4 can be chosen in a class of 0? Begin with finding the permutation: 0 19 18 17 Because order is not important in choosing a group, we must divide out the repeated groups. To divide out the repeated groups, divide by 4! 019 1817 41 4845 You Try: A club has 16 members. How many ways could a president, vice-president, and secretary be chosen by the members of the club. QOD: When given a group of elements, which would there be more of a permutation of n elements, or a combination of n elements? Eplain. Page 10 of 10 McDougal Littell: 11.1, 11., 11.4 11.6, 11.8