11-9 Rational Equations and Functions

Algebra 1 Glencoe McGraw-Hill Linda Stamper

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21

Rational equations are equations that contain rational expressions. You can use cross products to solve rational equations but only when both sides of the equation are single fractions.

od,c,b,a where dc

ba

Cross Product PropertyIf two ratios are equal then their cross products are

also equal.

8 8

Because division by zero is undefined, you must check your answers to make sure that any values of a variable that result in a zero denominator are excluded from the final answer.

23m

3m9m2

Solve.

Write the problem. 9m2 2

9m18m2 22

9m2 9m2

Any value for the variable that results in a zero in the denominator, is not a solution.

2m2m918m2

3m 3m

Cross multiply.Distribute.

3m3m

Example 1

x3

6xx

36xxx

183xx2 0183xx2

You must use parentheses!

18

36 3

03x6x 03x or 06x

3x or 6x

Check: Remember that division by zero is undefined, therefore any value that results in making the denominator zero is not a solution.

Solve. Example

2

41x

x3

1xx43

xx12 2 12xx0 2

12

1

4 3 3x4x0

03x or 04x 3x or 4x

Another method you can use to solve rational equations is to multiply each side of the equation by the LCD of all of the fractions on both sides of the equation. This will eliminate all of the fractions. This method works for any rational equation.

n1

6n3n

n2n

Remember: You can use cross products to solve rational equations but only when both sides of the equation are single fractions.

Multiply both sides by the LCD.

n1

6n3n

n2n

Distribute the LCD and simplify.

Solve.

n1

6n3n

n2n 6nn 6nn

n2n

16nn

2n6n 3nn 6n Multiply and solve.

6nn3n12n8n 22 6n12n5

612n6 18n6

3n

6n3n

16nn

16nn

n1

Example 3

2b3

22b5b2

Solve.

Example 4

c2c

282c

11 2

Example 3

2b3

22b5b2

Solve.

2b3

22b5b2

2b2b 2b2b

12

12b2b

2b5b2

12b2b

12b2b

2b3

5b22b 22b2b 2b3

10bb2 2 24b2 6b3

10bb2 2 8b2 2 6b3 8b210bb2 22 6b3

6b42 6b32b

b44 b1

2cc

2cc282c2cc

12cc

Example 4

Solve.

2cc

c2c

282c

11 2

2cc

2cc28

2c1

1

2cc28

2c1

1

28c2cc

28cc2c2 028c3c2

04c7c 04cor07c

4,7

2x

3x4x0

Recall that to find the roots of a quadratic function, find the values of x when y = 0. The roots of a rational function are found similarly.

Rational Functions

2x

12xxxf

2

2x12xx

02

Write the problem.

Set f(x) = 0.

Factor.

When x = 4 and -3, the numerator becomes zero, so f(x) = 0. Therefore, the roots of the function are 4 and -3.Check: Remember that division by zero is undefined, therefore any value that results in making the denominator zero is not a solution.

Example 5

Solve. Example

6

3x18x3x

xf2

2xx

8x6xxf 2

2

3x18x3x

02

3x

3x6x0

03xor06x

3xor6x

2xx

8x6x0 2

2

2xx

2x4x0 2

02xor04x

2xor4x

Check: Remember that division by zero is undefined, therefore any value that results in making the denominator zero is not a solution.

Example 7

Solve. Example

8

xx

2x1

1x5

2

Check: Remember that division by zero is undefined, therefore any value that results in making the denominator zero is not a solution.

53x

5x3x

Example 9

Example 10

1a2aa

1a

5aa

2

2

2

4x16x6x

xf2

7,0 43

8,2 3

11-A10 Page 630-632 # 9–26,45-50.