Algebra 1 Chapter 11 Notes Spragg Spring 010 Name: Algebra Homework: Chapter 11 (Homework is listed by date assigned; homework is due the following class period) HW# Date In Class Homework 33 Th 6/ Sections 11.1 and 11.: Simplifying Rational Expressions, Multiplying and Dividing Rational Expressions HW#33 Pg. 53: 14 even, 6, 44 Pg. 537: 9, 10, 4, 8, 9, 30 34 F 6/3 Section 11.3: Polynomial Long Division HW#34 Pg. 539: 7 10 Pg. 54:, 4, 9, 10, 17, 19, 4 35 M 6/6 Section 11.4: Adding and Subtracting Rational Expressions with Like Denominators HW#35 Pg. 54: 6, 13, 14, 16 Pg. 544: 57, 59 Pg. 547: even 36 T 6/7 Section 11.4: Adding and Subtracting Rational Expressions with Unlike Denominators HW#36 Pg. 547: 18 4 even, 30, 31, 38, 39 Pg. 560: 5 8, 17 1, 3 5 37 W 6/8 Section 11.5: Solving Rational Equations Work problems HW#37 Pg. 553: 1, 4, 5, 16, 17, 18,, 7, 8 Pg. 558: 31, 41, 43, 59 Print: Final Exam Packet Chapter 11 Test on Friday 38 Th 6/9 Chapter 11 Review HW#38 Chapter 11 Study Guide Correct Online Print: Final Exam Packet Chapter 11 Test on Friday 39 F 6/10 Chapter 11 Test HW#39 HW#39 Wkst: 1 9 all Print: Final Exam Packet Final on Thursday 6/16-1 -
- - Warm Up #33: 1.) Solve for x: x 3 x.) Find f( ) if f(x) = x 3 5 3.) Find the slope and y intercept of the line: 4x y = 6 4.) Simplify: (3x y 4 )( xy 3 ) 5.) Solve: 5x 1 14 6.) Solve for a and b: a b 9 3a b 17
Notes #33: Section 11.1: Simplifying Fractions - 3 - Simplifying Rational Expressions (Fractions): Factor all expressions (numerators and denominators) Reduce by canceling out (terms in parentheses) from the numerator and denominator Look to cancel opposites; leave a (-1) behind. Ex: 4 x = x 4 Re-write answer as a new and reduced fraction Simplify: 1.) 6 x 18 5x 15.) 4 x x x 3.) 3 y y y 4 4 y 3y 4.) 3 8k k 3 5k 0
Section 11.: Multiplying and Dividing Rational Expressions Multiplying/Dividing Rational Expressions: Factor all expressions (numerators and denominators) If a division problem, the second fraction and write as a problem. Reduce by canceling out (terms in parentheses) from the numerator and denominator [Look for and ways to!!!] Re-write answer as a new and reduced fraction - 4 - Ex: 4( x 5) (5 x) 9( x1) 3( x3) Simplify: 5.) 6 x 1 3 x 9 6.) 5x15 x4 x 4 x 1 3 x x x 7.) x 8 x x x x 4 10 8.) c 5c 3 c d c d c1 9.) 3 y y 4y 8 y 1 y 3y 3y y
Warm Up #34: Final Review 1.) Multiply. Leave your answer in scientific notation:.) Find f( 3) if f(x) = 4 x 4 11 6. x 10 3.1 x 10-5 - 3.) Are the lines parallel, perpendicular, or neither? (Hint: find the slope of each line) 4.) Solve: x 1 5 y 3x1 3xy 7 Notes #34: Section 11.3: Polynomial Long Division First, practice your long division skills: 1.) 3 0.85.) 8 341
Polynomial Long Division ( x 6x10) ( x ) 1.) Write the dividend and divisor in descending order; fill in all gaps with 0 s - 6 -.) Put the expression under the division sign 3.) Divide the terms only 4.) this number to the divisor, put below 5.) ; switch/switch to change to addition! 6.) Steps 3 5 again 7.) Put remainder over the divisor 3.) 5 3 14 8 6 x y x y xy xy 4.) 18rt 1rt 6rt 3 3rt 8 6 6 5 5 4 5.) (3x 3 + x 4x) (x + ) 6.) (a 3 + a 3) ( + a)
- 7-7.) (x 3 + 1) (x + 1) 8.) (6x 4 + 5x 3x 3 + x 6) (3x ) 9.) (3x 3 + 5x + 4 x) (4 + x) 10.) (m 3 + m + m + 1) (m + 1)
Warm Up #35: Final Review 1.) Multiply. Leave your answer in scientific notation: (.4 x 10 4 )(8.1 x 10 9 ).) Find f( 3) if f(x) = x x - 8-3.) Are these lines parallel, perpendicular, or neither? x4y 18 y x3 4.) Solve: 15x 7 x 5.) Find the equation of the line passing through the points (4, 1) and (, 6). Leave your answer in standard form. Notes #35: Section 11.4: Adding and Subtracting Rational Expressions Add/Subtract. Simplify. When adding/subtracting fractions, you need a Add/Subtract the numerators. Leave the denominators Factor and reduce 1.) 5 4.) 15 15 1 6z 6z 3.) 3bx cx ax ax x x 3 x 9 x 9 4.)
- 9-5.) a 3a a a 6.) 3 4x x x x 7.) 5 m m5 m5 8.) 8 y 3y 30 3y 30 9.) 3 n n 6 n4 n4 10.) 3x 8x 13x3x 4x14 4x 14 3ab ba 1.) 4ab 4ab 11.) 5 5 5x 7y 3xy 3xy
- 10 - **If the denominators do not match, you need to make them the same before you add/subtract** Find the Multiply each fraction by what its missing. Distribute in the numerator. Add/Subtract the numerators. Leave the denominators Factor the numerator and reduce 13.) 7 3 14.) x 4x 4 7x 5x 15.) 7 3z 5 16.) 6 4 8 6 5x 3x 7 9 17.) 18.) 3 n 7n 3 5 ab 3bc 19.) Use long division to evaluate: t 3t11 t 3
Warm Up #36: Final Review 1.) Divide. Leave your answer in scientific notation: 14 3.6 x 10 3 6.0 x 10.) Find h( 1) if h(x) = 6x 3-11 - 3.) Does the system have no solution, one solution, or infinitely many solutions? y x3 4xy6 4.) Simplify: 3 8m 40m 1m 60m 4 3 5.) Find the equation of the line with slope ½ and passing through the point ( 4, 7) 6.) Find the equation of the line passing through the points (3, 0) and (1, 10). Notes #36: Section 11.4: Adding and Subtracting Rational Expressions With Unlike Denominators Add/Subtract. Simplify. each denominator Find the Multiply each fraction by what its missing. Distribute in the numerator. Add/Subtract the numerators. Leave the denominators Factor the numerator and reduce 1.) 5 6z 3z.) 3 ax ay 3.) 3 5 x1x4 xx4
- 1-4.) 3 aa3 a3a4 5.) 3m 1 3 5 4 5 m m m 6.) x1 x x3 x5 7.) 5 3 6x y 9x3y 1 8.) 1 m m1 m 1 3 x x6 x 3x 9.)
- 13-10.) 7 5 5 6 x x x x 5x1 5x1 11.) 15 7 b 4 b 1 x 3 1.) x 5 x 4x5
- 14 - Warm Up #37: Final Review 1.) Find the vertex of the parabola: f(x) = x 8x + 1.) Solve using square roots: 6p 5 = 3.) Solve for y: 4y 3 8 4.) Simplify: 8 3 7 Notes #37: Section 11.5: Solving Rational Equations Factor denominators, find Set factored common denom = 0 and solve; these are the restrictions on the domain Multiply to make all denominators match Cancel out denominators; solve the left-over equation If you are left with a quadratic; solve by or Compare your solutions to the restrictions Solve: 4.) 1 1 1 x 5 LCM = x 5.) 6 x x 5 LCM = x
- 15-6.) 3 3 x4 x4 1 7.) 3 5 x x 7 3 1 1 x 1 x x x 8.)
- 16-1 9.) 1 4 y y y 4 Work/Job Problems Preview Questions: 10.) If you can paint a fence in 4 hours, what 11.) If you can write a paper in 3 hours, fraction of the job can you do in 1 hour? what fraction of the paper can you Meaning, what is your rate? write in 1 hour? Meaning, what is your rate? Work Problems: Fill in x = across the top of your chart Let t = time Rate = 1 hours it takes to complete a job Add jobs to = 1 whole job 1.) Laura can paint a fence in hrs. It takes her brother 3 hrs to paint the same size fence. If they worked together, how long would it take them to paint the fence? Laura Her brother Laura s job + her brother s job = 1 painted fence
- 17-13.) Christian can write his English essay in 3 hours. It takes his partner 4 hours to write it. If they worked together, how long would it take them to write an essay? Christian Partner 14.) Garden Hose A can fill a jacuzzi in 10 hours. Garden Hose B can fill a Jacuzzi in 1 hours. How long would it take both hoses to fill the tank? Hose A Hose B
Warm Up #38: Final Review 1.) Simplify: (4x y).) Divide: (x 3 5x + 10x 1) (x 3) - 18-4x 5 3.) Add: x 9 x3 5.) Find the equation of the line that is parallel to y = x + 1 and passing through the point (5, ) 4.) Graph: x 4y 7 y 10 9 8 7 6 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 x - -3-4 -5-6 -7-8 -9-10 6.) Find the equation of the line that is perpendicular to y = x + 1 and passing through the point (5, )
Notes #38: Chapter 11 Review - 19 - Chapter 11 Review For #1-4, solve: 1.) 3 1 m 3m 3.) x 3 x 4 x x 3.) 1 1 1 x1 x x 6 4.) x1 x3 x x3 Simplify: 5.) 4 4 y 3y 3 y y y 6.) 3 8k k 3 5k 0
Add/Subtract: 3 7.) x 5x6 x 4 1 1 3 6a ab 8b 8.) - 0 - Divide: 9.) 3x 7x 3x 3x 4 3 10.) (x 4 x 15) (x 3) 11.) Joe can paint a fence in 4 hours. His sister can paint the same fence in hours. How long would it take them to paint it if they worked together? Joe Sister
Chapter 11 Study Guide For #1-, simplify each expression 18 x 1.) x x4.) 3 x x x 5 5 x 5-1 - Name: For #3-4, multiply/divide 6x 13x6 6x x 3.) 4x 9 4x 1 6x 9 x1 4.) x 6x 4x 6 For #5-6, divide 4x 4 x 16 x 1 5.) 6.) 8x 3 y 4 1x y 3 36xy 4xy
For #7-8, add/subtract 3 4 7.) + x6 6x18 8.) 15 7 b 4 b - - 6 4 9.) x 3x x k 4 3 k1 10.) k 3k18 k3 k6 For #11-14, solve for the variable 1 b 11.) b 3 4 1.) 3 3n 4 3
- 3-13.) 1 1 4 y y y 4 14.) 4 5 x x3 For #15-16, complete the chart, write an equation, and solve. 15.) Sarah can paint a fence in 5 hours. Her sister can paint the same fence in hours. How long would it take them to paint it if they worked together? Sarah Sister 16.) Garden Hose A can fill a jacuzzi in 9 hours. Garden Hose B can fill a Jacuzzi in 6 hours. How long would it take both hoses to fill the tank? Hose A Hose B **Study Guide Ends Here; Please Stop!!**
- 4 - HW#39 Worksheet: 1 9 all Name: Simplify. Show ALL work. Leave answers in simplified form: y 5y7 1.).) 7 x 9 x 5 y 7 3x 3x 3.) 9 9 3 3 a a a a a a a 4.) x x x x x x x x 3 10 8 15 15 7 10 3x 1x 5.) x x 4 x 4 6.) x 8x15 x5
- 5-7.) Solve for x. List the LCM and the restrictions on x: x 8 x x x 1 1 1 LCM =, x 8.) Mrs. Spragg can wash and wax her car in hours. It takes Mr. Spragg 3 hours to do the same thing. If they were to work together, how long would it take them to do the job? a) Complete the chart: Mrs. Spragg Mr. Spragg b) Write an equation and solve: c) Write your solution: Divide: x 3 x10 x 3 9.) Divide: 10.) x x x 4 3 0
- 6-11.) Simplify: 3 5 x y x y xy x y 1.) x 1 3 x LCM =, x List the domain and range of each relation. Is the relation a function? 13.) {(-, 1), (0, 1), (5, 1), (3, 0)} 14.) {(4, -), (0, ), (4, 3), (5, 1)} Is the relation a function? 15.) Domain Range 16.) Domain Range
17.) 18.) y 10 9 8 7 6 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 x - -3-4 -5-6 -7-8 -9-10 y 10 9 8 7 6 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 x - -3-4 -5-6 -7-8 -9-10 - 7 - Find the indicated outputs for these functions: 19) f(x) = -x + 5; find f(0), f(-) and f(3) Find and label the vertex, axis of symmetry, and x-intercepts. List the domain and range of the quadratic function. 0.) f(x)= x + 10x + 1 y 10 9 8 7 6 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8 9 10 x -1 - -3-4 -5-6 -7-8 -9-10 1.) h(x) = x + 10x 16 Vertex: X Intercepts: Axis of Symmetry: Domain: Range: y 10 9 8 7 6 5 4 3 1-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 x - -3-4 -5-6 -7-8 -9-10 Vertex: X Intercepts: Axis of Symmetry: Domain: Range:
Solve by factoring:.) x + 1x = -10 3.) 16(x 1) = x(x + 8) - 8 - Solve by using square roots: 4.) 3b 1 = 7 5.) (3x + 1) = 5 Solve by completing the square: 6.) x 10x 11 = 0 7.) x 4x 6 = 0 Solve by using the quadratic formula: 8.) 3x x = 8 9.) x = 9 6x Don t forget to go online and print your Final Exam Study Guide!!