11-9 Rational Equations and Functions
Algebra 1 Glencoe McGraw-Hill Linda Stamper
84
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.