1. General Rational Equations

2. General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.

3. This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.

4. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.

5. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
Let A = apple, B = banana, and C = cantaloupe

6. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
Let A = apple, B = banana, and C = cantaloupe
The relation is
3
4
A
1
2
B +=
1
3
C

7. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
1
3
C
Let A = apple, B = banana, and C = cantaloupe
The relation is

8. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12
Let A = apple, B = banana, and C = cantaloupe
The relation is

9. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12 Distribute it to each term.
Let A = apple, B = banana, and C = cantaloupe
The relation is

10. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12 Distribute it to each term.
3
Let A = apple, B = banana, and C = cantaloupe
The relation is

11. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12 Distribute it to each term.
3 6 4
Let A = apple, B = banana, and C = cantaloupe
The relation is

12. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12 Distribute it to each term.
3 6 4
9A = 6B + 4C
Let A = apple, B = banana, and C = cantaloupe
The relation is

13. Example A. Suppose the value of 3/4 of an apple is the same
as the value of 1/2 of a banana with 1/3 of a cantaloupe,
restate this relation in whole values.
This is accomplished by multiplying the LCD of all the
fractional terms in the equation, to each term on both sides,
then clear each denominator using these multiplications.
General Rational Equations
An equation or relation expressed using fractions may
always be restated in a way without using fractions.
The LCD =12, multiply it to both sides.
3
4
A
1
2
B +=
3
4 A
1
2 B +=( )
1
3
C
1
3 C * 12 Distribute it to each term.
3 6 4
9A = 6B + 4C
Hence the value of 9 apples is the same as
6 bananas and 4 cantaloupes.
Let A = apple, B = banana, and C = cantaloupe
The relation is

14. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.

15. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2

16. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)

17. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)

18. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x + 2)

19. General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)

20. 3(x + 2) = 6( x – 1)
General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)

21. 3(x + 2) = 6( x – 1)
3x + 6 = 6x – 6
General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)

22. 3(x + 2) = 6( x – 1)
3x + 6 = 6x – 6
6 + 6 = 6x – 3x
General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)

23. 3(x + 2) = 6( x – 1)
3x + 6 = 6x – 6
6 + 6 = 6x – 3x
12 = 3x
4 = x
General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)

24. 3(x + 2) = 6( x – 1)
3x + 6 = 6x – 6
6 + 6 = 6x – 3x
12 = 3x
4 = x
General Rational Equations
To solve an equation with fractional terms, we first clear the
fractions by multiplying both sides by the LCD.
Example B. Solve
3
x – 1 =
6
x + 2
Multiply both sides by the LCD = (x – 1)(x + 2)
3
x – 1
=
6
x + 2
( ) * (x – 1)(x + 2)
(x – 1)(x + 2)
This is a proportional equation.
Multiplying the LCD yield the same simplified equation
if we cross-multiplied.

25. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1 + 1

26. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
+ 1

27. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1

28. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x + 1)

29. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1)

30. General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)

31. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)

32. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2

33. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10

34. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12

35. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12
0 = (x + 4)(x – 3) hence x = –4, 3

36. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12
0 = (x + 4)(x – 3) hence x = –4, 3
However, this method of clearing the denominator might
produce a solution(s) that does not work for the original
fractional equation.

37. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12
0 = (x + 4)(x – 3) hence x = –4, 3
However, this method of clearing the denominator might
produce a solution(s) that does not work for the original
fractional equation. Specifically, we have to check that the
answers obtained will not turn the denominator into 0 in the
original problem.

38. 2(x + 1) = 4(x – 2) + (x – 2)(x + 1)
General Rational Equations
Example C. Solve
2
x – 2
=
4
x + 1
Multiply both sides by the LCD : (x – 2)(x + 1)
2
x – 2 = 4
x + 1
( ) * (x – 2)(x + 1)
+ 1
+ 1
(x – 2)(x + 1) (x – 2)(x + 1)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12
0 = (x + 4)(x – 3) hence x = –4, 3
However, this method of clearing the denominator might
produce a solution(s) that does not work for the original
fractional equation. Specifically, we have to check that the
answers obtained will not turn the denominator into 0 in the
original problem. In this example, both x = –4, 3 are good
answers because they don’t turn the denominators to 0.

39. General Rational Equations
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.

40. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.

41. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.
2
x – 2
=
4
x + 1 +
Treat the 1 as .
1
1
1
1

42. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.
2
x – 2
=
4
x + 1 +
Treat the 1 as .
1
1
1
1
2
x – 2
=
4 + (x + 1)
x + 1

43. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.
2
x – 2
=
4
x + 1 +
Treat the 1 as .
1
1
1
1
2
x – 2
=
4 + (x + 1)
x + 1
2
x – 2
=
x + 5
x + 1

44. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.
2
x – 2
=
4
x + 1 +
Treat the 1 as .
1
1
1
1
2
x – 2
=
4 + (x + 1)
x + 1
2
x – 2
=
x + 5
x + 1

45. General Rational Equations
Example D. Solve 2
x – 2 = 4
x + 1 + 1
We may use the cross multiplication to combine the two
terms on the same side first to arrange the problem as a
proportional problem.
2
x – 2
=
4
x + 1 +
Treat the 1 as .
1
1
1
1
2
x – 2
=
4 + (x + 1)
x + 1
2
x – 2
=
x + 5
x + 1
2(x + 1) = (x – 2)(x +5)
You finish it.
.
.

46. General Rational Equations
An answer that doesn’t work for the original problem is called
an extraneous solution.

47. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.

48. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.

49. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)

50. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1

51. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1

52. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)

53. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)

54. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)
3 = –x + 6

55. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)
3 = –x + 6
x = –3 + 6

56. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)
3 = –x + 6
x = –3 + 6
x = 3

57. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)
3 = –x + 6
x = –3 + 6
x = 3
However this is an extraneous answer because it is not
usable – it turns the denominator into 0.

58. General Rational Equations
Example E. (Extraneous solution) Solve 3
x – 3
=
x
x – 3
– 2
An answer that doesn’t work for the original problem is called
an extraneous solution.
Multiply the LCD (x – 3) to both sides.
3
x – 3
=
x
x – 3
– 2( ) (x – 3)
1 1 (x – 3)
3 = x – 2(x – 3)
3 = –x + 6
x = –3 + 6
x = 3
However this is an extraneous answer because it is not
usable – it turns the denominator into 0.
Hence there is no solution for this rational equation.

59. General Rational Equations
Ex. A. Rewrite the following fractional equations into ones
without fractions. You don’t have to solve for anything.
1
4
A
1
2
B +=
2
3
C1.
5
6
A
3
4
B =–
2
3
C2.
5
4
A
6
5
B –= 2C3.
1
6
A 3B =–
2
7
C4.
1
x
1
y
=+5.
1
x
1
y
= 1–6.
1
xy
1
x
1
y
=+7.
1
(x – y)
= x– 18.
1
z
Ex. B. Rewrite the following equations into linear equations
without fractions. Solve for the x and check the answers.
2
x =+ 19.
3
x = 110.1
2x
1
3x–
3
4x
=– 111.
6
7x
= 112.5
6x
7
8x
–

60. General Rational Equations
Ex. C. Rewrite the following equations into linear equations
without fractions. Solve for the x and check the answers.
2
x – 1
=+ 113. 3
2x – 2
3
x – 2
=+ 214.
1
3x – 6
5
x – 2
=– 215.
1
6 – 3x
3
x =
3
x + 2 + 2
Ex. D. Rewrite the following into 2nd degree equations.
Solve for the x and check the answers.
20.
3
x – 2
–
3
x + 2
= 1
21.
2
x
= x+ 116.
3
x
= x – 217.
–6
x
= x+ 518.
–3
x
= x – 419.
6
x + 1 =
6
x – 1
22.
12
x – 1
–
12
x + 1
= 123.

61. General Rational Equations
24. 1
x
+
1
x – 1
=
5
6
25. 1
x
+
1
x + 2
=
3
4
26. 1
x + 1
x – 2 = 4
3
27. 1
x + 2
x + 2 = 1
28. 2
x
+
1
x + 1
=
3
2
29. +
5
x + 2
= 2
2
x – 1
30. –
1
x + 1
=
3
2
31.6
x + 2
–
4
x + 1
= 1
1
x – 2