5.6 solve rational equations

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5.6 Solve Rational 5.6 Solve Rational Equations Equations p. 345 p. 345 What is the easiest way to solve a What is the easiest way to solve a rational equation? rational equation? What is an extraneous solution? What is an extraneous solution? How do you know when to cross How do you know when to cross multiply or multiply out the multiply or multiply out the denominator? denominator?

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5.6 Solve Rational Equations. p. 345 What is the easiest way to solve a rational equation? What is an extraneous solution? How do you know when to cross multiply or multiply out the denominator?. Cross Multiplying. - PowerPoint PPT Presentation

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Page 1: 5.6  Solve Rational Equations

5.6 Solve Rational Equations5.6 Solve Rational Equationsp. 345p. 345

What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?

What is an extraneous solution?What is an extraneous solution?

How do you know when to cross multiply or How do you know when to cross multiply or multiply out the denominator?multiply out the denominator?

Page 2: 5.6  Solve Rational Equations

Cross Multiplying

When there is only one fraction on each side of the =, When there is only one fraction on each side of the =, just cross multiply as if you are solving a proportion.just cross multiply as if you are solving a proportion.

Beware of extraneous solutions!Beware of extraneous solutions!

Page 3: 5.6  Solve Rational Equations

Solve: 3x + 1=

94x + 1

3x + 1=

94x + 1 Write original equation.

3(4x + 5) = 9(x + 1) Cross multiply.12x + 15 = 9x + 9 Distributive property3x + 15 = 9 Subtract 9x from each side.

3x = – 6 Subtract 15 from each side.x = – 2 Divide each side by 3.

The solution is –2. Check this in the original equation.

ANSWER

Page 4: 5.6  Solve Rational Equations

Example: Solve.

4

1

4

32

xxx

xx 42 12342 xxx

0122 xx0)3)(4( xx

03or 04 xx3or 4 xx

Check your solutions!Check your solutions!

)4(3 x

Page 5: 5.6  Solve Rational Equations

Solve 1

2

22

62

x

x

xx

)1(6 x)2)(1(2 xxx

6)2(2 xx3)2( xx

0322 xx0)1)(3( xx

01or 03 xx1or 3 xx

1

2

)1(2

6

x

x

xx

Check your solutions!Check your solutions!

Page 6: 5.6  Solve Rational Equations

An alloy is formed by mixing two or more metals. Sterling silver is an alloy composed of 92.5% silver and 7.5% copper by weight. Jewelry silver is composed of 80% silver and 20% copper by weight. How much pure silver should you mix with 15 ounces of jewelry silver to make sterling silver?

ALLOYS

SOLUTION

Percent of copper in mixture =Weight of copper in mixture

Total weight of mixture

Page 345

Page 7: 5.6  Solve Rational Equations

7.5100

0.2(15)15 + x= x is the amount of silver added.

7.5(15 + x) = 100(0.2)(15) Cross multiply.

112.5 + 7.5x = 300 Simplify.7.5x = 187.5 Subtract 112.5 from each side.

x = 25 Divide each side by 7.5.

You should mix 25 ounces of pure silver with the jewelry silver.

ANSWER

Page 8: 5.6  Solve Rational Equations

Hint for solving:Hint for solving:

• A quick way to solve a rational A quick way to solve a rational equation is to multiply everything equation is to multiply everything through by the LCD. through by the LCD.

• This will get rid of all the fractions!This will get rid of all the fractions!

• Beware of extraneous solutions!Beware of extraneous solutions!

Page 9: 5.6  Solve Rational Equations

Example: Solve.

xx

12

2

13

LCD: 2x

Multiply each fraction through by the LCD

x

xx

x

x 12*2

2

23*2

246 x

18 x

18xCheck your solution!Check your solution!

18

12

2

1

18

3

1293

Page 10: 5.6  Solve Rational Equations

Solve: 1 – 8x – 5

=3x

1 – 8x – 5

=3x

Write original equation.

x(x – 5) 1– 8x – 5( ) = x(x – 5) 3

xMultiply each side by the LCD, x(x–5).

x(x –5) – 8x = 3(x – 5) Simplify.

x2 – 5x – 8x = 3x – 15 Simplify.

x2 – 16x +15 = 0 Write in standard form.

(x – 1)(x – 15) = 0Factor.

x = 1 or x = 15 Zero product property

The solutions are 1 and 15. Check these in the original equation.

ANSWER

Page 11: 5.6  Solve Rational Equations

Solve.1

54

1

5

xx

x LCD: ?LCD: (x+1)

)1(

)1(5)1(4

)1(

)1(5

x

xx

x

xx

5445 xx

145 xx

1x

Check your solution!Check your solution!

11

54

11

)1(5

0

54

0

5

?

No Solution!No Solution!

Page 12: 5.6  Solve Rational Equations

Solve. 14

6

2

232

xx

x

Factor 1st!

1)2)(2(

6

2

23

xxx

x

LCD: (x+2)(x-2)

)2)(2()2)(2(

)2)(2(6

)2(

)2)(2)(23(

xx

xx

xx

x

xxx

42264263 22 xxxxxx

2443 22 xxx

0642 2 xx0322 xx

0)1)(3( xx

01or 03 xx

1or 3 xx

Check your solutions!Check your solutions!

Page 13: 5.6  Solve Rational Equations

• What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?

Multiply the equation by a value that will Multiply the equation by a value that will make the denominator equal to 1.make the denominator equal to 1.

• What is an extraneous solution?What is an extraneous solution?

A solution that does not make a true A solution that does not make a true statement in the original equation.statement in the original equation.

• How do you know when to cross multiply How do you know when to cross multiply or multiply out the denominator?or multiply out the denominator?

Cross multiply if there is only one fraction Cross multiply if there is only one fraction on each side of the equal sign, otherwise on each side of the equal sign, otherwise multiply out the denominator.multiply out the denominator.

Page 14: 5.6  Solve Rational Equations

5.6 Assignment

p. 348

5-27 odd