Transcription of CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS …
1 CHAPTER 11 295 CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS CHAPTER Objectives By the end of this CHAPTER , students should be able to: Identify extraneous values Apply methods of solving RATIONAL EQUATIONS to solve RATIONAL EQUATIONS Solve APPLICATIONS with RATIONAL EQUATIONS including revenue, distance, and work-rate problems Contents CHAPTER 11: RATIONAL EQUATIONS AND APPLICATIONS .. 295 SECTION : RATIONAL EQUATIONS .. 296 A. EXCLUDED VALUES REVIEW .. 296 B. SOLVE RATIONAL EQUATIONS BY CLEARING denominators WITH THE LCD.
2 297 C. FACTORING denominators .. 299 D. SOLVING RATIONAL EQUATIONS WITH EXTRANEOUS SOLUTIONS .. 300 EXERCISE .. 302 SECTION : WORK-RATE PROBLEMS .. 303 A. ONE UNKNOWN TIME .. 303 B. TWO UNKNOWN TIMES .. 305 EXERCISE .. 307 SECTION : UNIFORM MOTION PROBLEMS .. 308 A. UNIFORM MOTION PROBLEMS .. 308 B. UNIFORM MOTION PROBLEMS WITH STREAMS AND WINDS .. 310 EXERCISE .. 312 SECTION : REVENUE PROBLEMS .. 313 EXERCISE .. 315 CHAPTER REVIEW .. 316 CHAPTER 11 296 SECTION : RATIONAL EQUATIONS When solving RATIONAL EQUATIONS , we can solve by using the same strategy we used to solve linear EQUATIONS with fractions : clearing denominators .
3 However, we first need to revisit excluded values. A. EXCLUDED VALUES REVIEW Note A RATIONAL expression is undefined where the denominator is zero. MEDIA LESSON Find excluded values of a RATIONAL expression (Duration 4:20) View the video lesson, take notes and complete the problems below Find values where a RATIONAL expression is undefined. a) 174 32 b) +19( 10)( 4) c) 9 +10 2 9 10 CHAPTER 11 297 Definition Recall, the excluded values are values which make the expression undefined.
4 Hence, when solving a RATIONAL equation, the solution(s) is any value(s) except the excluded values. If we obtain a solution that is an excluded value, we call this an extraneous solution. B. SOLVE RATIONAL EQUATIONS BY CLEARING denominators WITH THE LCD Steps for solving RATIONAL EQUATIONS with the same denominator Step 1. Determine the excluded values of the equation. Step 2. Clear denominators by multiplying each term by the lowest common denominator. Step 3. Solve the equation. Step 4. Verify that the solutions obtained are not an excluded value.
5 MEDIA LESSON Solve RATIONAL EQUATIONS by clearing the denominators -part 1 (Duration 9:15) View the video lesson, take notes and complete the problems below Example: Solve the RATIONAL EQUATIONS . a) +86+3=3 +88 6: _____ 8:_____ LCD: _____ Excluded values: _____ b) 4 2+12=34 LCD=_____ Excluded values: _____ YOU TRY a) Solve for : 23 56=34 b) Solve for : 9 +1 52=43 CHAPTER 11 298 MEDIA LESSON Solve RATIONAL EQUATIONS by clearing the denominators - part 2 (Duration 4.
6 32) View the video lesson, take notes and complete the problems below Example: Solve the RATIONAL EQUATIONS . c) 25 3= 4 ____ d) 2 16 =4 ____ YOU TRY a) Solve for : 5 +5 +2 +3 = 2 +2 b) Solve for : +2+1 +1=5( +1)( +2) CHAPTER 11 299 C.
7 FACTORING denominators Often we will need to factor denominators before finding the LCD. MEDIA LESSON Solve RATIONAL EQUATIONS with factoring the denominators first- Part 1 (Duration 4:27) View the video lesson, take notes and complete the problems below Example: Solve the RATIONAL EQUATIONS . a) 2+ 2 4= +3 +2 Excluded values: _____ MEDIA LESSON Solve RATIONAL EQUATIONS with factoring the denominators first Part 2 (Duration 4:45) View the video lesson, take notes and complete the problems below Example: Solve the RATIONAL EQUATIONS .
8 B) 3 +6+ 2 3= 2 2+3 18 Excluded values: _____ CHAPTER 11 300 YOU TRY a) Solve for : 1 1 2=11 2 3 +2 b) Solve for : 2 +1+3 +1= 2 2+2 +1 D. SOLVING RATIONAL EQUATIONS WITH EXTRANEOUS SOLUTIONS MEDIA LESSON Solve a RATIONAL equation with no solution (Duration 5:07 ) View the video lesson, take notes and complete the problems below Solve the RATIONAL EQUATIONS .
9 A) 6 36 9= 1 6 Excluded values: _____ CHAPTER 11 301 MEDIA LESSON Solve a RATIONAL equation with extraneous solutions (Duration 5:00 ) View the video lesson, take notes and complete the problems below RATIONAL EQUATIONS extraneous Because we are working with fractions , the _____ cannot be _____. Solve the RATIONAL EQUATIONS . b) 8 2 4= 3 +56 2 12 +32 c) 2+2 4=4 12 2 6 +8 YOU TRY a) Solve for : +5 2 9= 11 +15 2 4 45 CHAPTER 11 302 EXERCISE Solve.
10 Be sure to verify all solutions. 1) 3 12 1 =0 2) + 20 4=5 4 2 3) + 6 3=2 3 4) 2 3 4=4 +56 1 33 4 5) 3 2 5 73 +1=32 6) 4 1 =123 7) 73 +12=34 8) 23 68 =1 9) +1= 4 +1 10) 4 1=12 3 +1 11) 4 2 6 45 15=12 12) 1 3+ +2 +3=34 13) 2 +3 1 2=1 2+ 6 14) 3 55 5+5 17 7 41 =2 15) 5 9+ +3 3= 4 2 2 12 +27 16) 2 +1 3 +5= 8 2 2+6 +5 17) 2 +2+2 4=3 2 2 8 18) +23 1 1 =3 33 2 19) 6 +52 2 2 +2 2 1=3 2 1 20)
