5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 To solve...

$: 5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 To solve an equation with fractions Procedure: To solve a fractional.$
1

5.5 Solving Fractional Equations

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

To solve an equation with fractions

Procedure: To solve a fractional equation

Step 3. Determine LCM of denominators.

Step 4. Multiply all terms by LCD (clearing denominators).

Step 5. Solve equation as in previous lessons.

Step 2. Determine any restrictions for the fractional equation.

Step 6. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step.

Step 1. Factor all denominators.

Step 7. Check

2



2a 1 a 1 SolvExamp e

9le 1

6 6.

9

Solution:

1. Find the LCDLCD=18

2. Multiply all fractions by the LCD.

2a 1 a 19 6

(18) (18) (18) (18)

9 6

3. Cancel the denominators.2 2 33

4. Simplify and solve.

4a 3 2a 3

2a 6Answer: a 3

5. Check.

2(3) 1 3 19 6 9 6

12 318 18

6 318 18

9 918 18

3 19 6

6 19 6

Your Turn Problem #1

x 5 1 x 2 1Solve and Check:

9 3 4 4

Answer: x = 7

3



Rational Expressions have restrictions. Restrictions are values for the variable which would make the fraction undefined.

x 5 3 3 2x

, Examples: , x 5 4x

x 0

In this section, we will have fractions where the denominator may contain the variable. If the denominator contains a variable, the equation is said to have restrictions.

Procedure: To solve a fractional equation (for current section)

Step 2. Determine LCM of denominators (LCD).

Step 3. Multiply all terms by LCD (clearing denominators).

Step 4. Solve equation as in previous lessons.

Step 1. Determine any restrictions for the fractional equation.

Step 5. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step.

Step 6. Check all answers which are not restrictions.

4



9 1 5 Solve

4xExamp

3le 2.

2x

Solution:

1. State restrictions

LCD=12x

2. Find LCD.

(12x) (12x) (9 1 54x 3

)

2x

12x

3. Multiply all terms by LCD.3 6 4


27 4x 30 5. Answer not equal to restriction.

x 0

6. Check. 9 1 53 334 24 4

9 1 5 33 32

10 103 3

4x 3

3Answer: x

4


5 3 7Solve and Check:

4x 2 5x

1Answer:

10

5



1 10 Solve and Check xExample 3

x 3.

Solution:


LCD=3x

2. Find LCD.

(3x) (3x 1 10x

x 3

) (3x) 3. Multiply all terms by LCD.

4. Simplify and solve.23x 3 10x

23x 10x 3 0

(3x 1)(x 3) 0

x 0

3x 1 0 or x 3 0

5. Answer not equal to restriction.

6. Check: (Not Shown)

1Answer: x , 3

3


1 37Solve and Check: x

x 6

1Answer: , 6

6

6



3 2 Solve Exa

xm

1 x 5ple 4.

Solution:


2. Find LCD.

3. Multiply all terms by LCD.


3x 15 2x 2 5. Answer not equal to restriction.

x 5, 1

LCD (x 1)(x 5) 3 2

x

(x 1)(x 5) (x 1)(x

1 x 5

5)

6. Check (not shown).Answer: x 17

Your Turn Problem #42 4

Solve and Check: 2x 1 x 2

4Answer: x

3

This example was solved by multiplying both sides of the equation by the LCD. Cross multiplication is another method for solving fractional equations where there is only one fraction on each side. a c

If , b 0, d 0 , then ad=bc.b d

3 2x 1 x 5

Using the Cross-multiplication property:

3(x 5) 2(x 1) 3x 15 2x 2

x 17 Next Slide

7



x 8 Solve 1

xExample

15

2 .

x

Solution:


2. Find LCD.



2 2x x 1(x 3x 2) 8x 16

x 1, 2

LCD (x 2)(x 1)

(x 2)(x 1) (x 2)(x 1) (x 2)(x 8 x 1)1

x 2 x 1

5. Answer not equal to restriction.

6. Check.

3 81

3 2 3 1

4 4

3 81

1 2

22x 4x 2 8x 16

Answer: x 3

22x 12x 18 0 22(x 6x 9) 0

2(x 3)(x 3) 0


x 1 2Solve and Check:

x 1 2 x 2

2Answer: , 3

3

8



Proportion Application Problems

To recognize a problem is a proportion problem, a sentence will be given which can be written as a fraction.

Examples: 3 out of 20 smoke; 4 panels generate 1500 watts;4

1500320

Procedure: To solve a proportion problem

2. Find the recognizable ratio and write it as a fraction.

1. Define the variable (what you are looking for?)

3. After the fraction is written write an equal sign. Then write a fraction using the remaining information with the defined variable. Make sure the fraction is written so both of the numerators have the same units and the denominators have the same units.

Examples: lbs lbs $ $

watts watts panels panels

,

4. Solve and answer the question.

9



Example 6. A chef estimates that 50 lbs of vegetables will serve 130 people. Using this estimate, how many pounds will be necessary to serve 156 people?

Solution:Let x = number of pounds of vegetables for 156 people.

lbs lbs people people

Make units match:

50 lbs x 130 people 156 people

Equation:

Solve by cross multiplying:

130x (50)(156)

130x 7800

x 60Answer: 60 lbs of vegetables are needed


A carpenter estimates that he uses 8 studs for every 10 linear feet of framed walls. At this same rate, how many studs are needed for a garage with 35 linear feet of framed walls?

Answer: 28 studs are needed

10



Let x = federal income tax on $125,000

12,500 taxes x taxes 50,000 income 125,000 income

taxes taxes income income

Make units match:

Reduce, then solve by cross multiplying:

4x 125,000

x 31,250

Answer: The federal income tax will be $31,250.

1

4

Example 7: The federal income tax on $50,000 of income is $12,500. At this rate, what is the federal income tax on

$125,000 of income? Solution:


If a home valued at $120,000 is assessed $2160 in real estate taxes, then how much, at the same rate, are the taxes on a home valued at $200,000?

Answer: The property would be $3600.

11



Example 8. A sum of $1400 is to be divided between two people in the ratio of 3 to 5. How much does each person receive?

x 31400 x 5

1st person receives ratio given

2nd person recieves

Solve by cross multiplying: 5x 3(1400 x)

5x 4,200 3x

The first person will receives $525, Answer: and the second person will receive $875.

Solution: Let x = amount first person receives

Let 1400 – x = amount second person receives

x 525

525 = amount for first person.To find amount for second person, 1400 – 525 = 875.

A sum of $5000 is to be divided between two people in the ratio of 3 to 7. How much does each person receive?


Answer: The first person receives $1500 and the second person receives $3500.

12



2 2

4 x 1 Solve and Example Check

x 9 x9.

3x2. State restrictions

3. Find LCD.

4. Multiply all terms by LCM.


24x x 2x 3

x 1 or x 3

6. Since x can not equal 3, cross it out as a solution.

x 3,0,3

1. Factor Denominators

4 x 1(x 3)(x 3) x(x 3)

LCD x(x 3)(x 3)

x(x 3)(x 3) x(x 3)(x 3) ( )

20 x 2x 3

0 (x 1)(x 3)

7. Check x = 1 (not shown).

Answer: x 1


2 2

2 x 3Solve and Check:

x 16 x 4x

Answer: x 3

13



2

x 2 63 Solve Example 10.

x 4 x 8 x 4x

32

Solution: 2. State restrictions

3. Find LCD.



2x 8x 2x 8 63

6. Answers not equal to restrictions.

x 8,4

1. Factor Denominators

x 2 63

x 4 x 8 (x 8)(x 4)

LCD (x 4)(x 8)

2x 6x 55 0

(x 11)(x 5) 0

7. Check both –11 and 5 (not shown).

Answer: 11,5

(x 4)(x 8) (x 4)(x 8) (x 4)(x 8)


2

5y 4 2 5Solve and Check:

2y 3 3y 46y y 12

Answer: y 1

14



Example: If a carpenter can build a shed in 8 hours, then his

rate of work is of the task completed per hour.18

Also, if this carpenter works for 5 hours, then his part of task completed is

1 5 tasks per hour 5 hours of a task completed.

8 8

Work problems are common in algebra because of the use of the rate concept as well as their application to real-life situations.

Work Problems

Rate of work: The portion of a task completed per one unit of time.

Important: In Work Problems, the time it takes a person (or machine) to do a task is the reciprocal of his/her rate!

Basic Work Formula: (Rate of work) (Time Worked) = (Part of Task completed)In most work problems, there will always be two individuals or devices who will contribute to the completion of the task.The formula for work problems will be to add the “part of task completed” for each individual or device and set it equal to “1” (for 1 completed job).

part of task completed(individual #1)

part of task completed(individual #2)

1Next Slide

15



Procedure: To Solve Work Problems

Step 1. Write down the “let statements” (what you are looking for) and any information given in the problem.Step 2. Create a chart with columns “Rate of work”, “Time worked”, and “Part of task completed.”Step 3. To find rate of work, make a fraction with a numerator equal to 1; and a denominator equal to the time it would have taken the individual or device to complete the task alone.

Step 4. To find time worked, write in the actual time the individual or device worked or will work on the task.

Step 5. To find Part of task completed, use the formula: (rate of work) (time worked) = (part of task completed)

Step 6. To write equation, add the “part of task completed” and set equal to 1.Step 7. Solve and answer question.

Next Slide

16



Example 11. A carpenter can build a shed working alone in 5 hours. His assistant working alone would take 10 hours to build a similar shed. If the two work together, how long will it take them to build the shed?

x x1

5 10

Let x = time to build shed togetherCarpenter: 5hrs., Assistant: 10 hrs.

1. Define variable (Let statements)

2. Create chart.

x5

x10

x

x

2x x 10

assistant

carpenter

Rate of work

Time worked

Part of Task

completed

3. Write in rate of work for each.

15

110

4. Write in time worked for each.

5. Write in part of task completedby multiplying rate time.

6. Obtain equation by adding the column and setting it equal to 1.

7. Solve and answer question.

1x 3

3

1Answer: It will take the carpenter and the assistant 3 hours.

3


Maria can do a certain job in 50 minutes, whereas it takes Kristin 75 minutes to do the same job. How long would it take then to do the job together?Answer: It would take them 30 minutes to do the job together.

(Multiply all terms by LCD)

The End.B.R. 12-30-06

5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 To solve...

Documents

Transcript of 5.5 Solving Fractional Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 To solve...