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Rational equations:
- equations containing one or more rational expressions
o remember from Lesson 7 that rational expressions are simply
fractions containing polynomials, so each equation we work
with today will contain fractions
o Example:
๐ฅ+ 4
5 ๐ฅโ 3
2 ๐ฅโ 5
10 ๐ฅ+ 7
- since we are working with equations now, rather than expressions, we
do NOT have to work with the fractions; instead we can eliminate
them using multiplication
o in the case of
๐ฅ+ 4
5 ๐ฅโ 3
2 ๐ฅโ 5
10 ๐ฅ+ 7
= 0 we could multiply both sides of
the equation by 5 ๐ฅ โ 3 and 10 ๐ฅ + 7 to eliminate the fractions
o if you prefer to get a common denominator and work with the
fractions, you may, but I do not recommend it
- keep in mind that even if we eliminate the fractions, we still cannot
have any answers that result in a denominator of zero
o in the case of
๐ฅ+ 4
5 ๐ฅโ 3
2 ๐ฅโ 5
10 ๐ฅ+ 7
3
5
or ๐ฅ โ โ
7
10
3
5
7
10
o the answer to
๐ฅ+ 4
5 ๐ฅโ 3
2 ๐ฅโ 5
10 ๐ฅ+ 7
= 0 is ๐ฅ = โ
1
6
, and since โ
1
6
does
not result in a denominator or zero, it is a valid answer
- ALWAYS exclude any values that make the denominator zero
- ALWAYS make sure that your solution(s) is/are not excluded
Steps for Solving Rational Equations:
- factor all denominators
- multiply both sides of the equation by the least common multiple of
all the denominators to eliminate the fractions
- distribute and combine like terms
- isolate the variable
- verify that your solution(s) is/are not restricted
Example 1: Solve the following equation. If there is no solution, write
NO SOLUTION. If there are infinitely many solutions, list the restrictions
using the notation ๐
โ
(all real numbers except).
2
a.
Example 3: Solve the following equation. If there is no solution, write
NO SOLUTION. If there are infinitely many solutions, list the restrictions
using the notation ๐
โ
(all real numbers except).
2
Be sure to ALWAYS check your answers in the original equation to be
sure they do not result in a denominator of zero. If they do, they are not
valid answers. If you only have one solution, and it is not valid, then you
now have NO SOLUTION. No solution does not mean we didnโt come
up with any answers (on Example 3 we did, ๐ฅ = โ 2 ); it means the answer
we came up with is invalid because it makes a denominator equal to zero.
Example 4 : Solve the following equationS. If there is no solution, write
NO SOLUTION. If there are infinitely many solutions, list the restrictions
using the notation ๐
โ
(all real numbers except).
a. 17 โ
3
๐ฅ
= โ 16 b.
3 ๐ฅ+ 1
6 ๐ฅโ 1
2 ๐ฅ+ 5
4 ๐ฅโ 13
3
๐ฅ
= โ 16 b.
3 ๐ฅ+ 1
6 ๐ฅโ 1
2 ๐ฅ+ 5
4 ๐ฅโ 13
b. B.๐ฅ(
3
๐ฅ
3 ๐ฅ+ 1
6 ๐ฅโ 1
) = (
2 ๐ฅ+ 5
4 ๐ฅโ 13
) ( 4 ๐ฅ โ 13 )( 6 ๐ฅ โ 1 )
2
2
3
33
2
2
๐
๐๐
๐
๐๐
c.
5
2 ๐ฅ+ 3
4
2 ๐ฅโ 3
14 ๐ฅ+ 3
4 ๐ฅ
2
โ 9
d.
6
๐ฅ+ 3
5
๐ฅโ 2
โ 20
๐ฅ
2
+๐ฅโ 6
5
2 ๐ฅ+ 3
4
2 ๐ฅโ 3
14 ๐ฅ+ 3
4 ๐ฅ
2
โ 9
d.
6
๐ฅ+ 3
5
๐ฅโ 2
โ 20
๐ฅ
2
+๐ฅโ 6
d.
Answers to Examples:
3. ๐๐ ๐๐๐ฟ๐๐๐ผ๐๐ ; 4 a.
1
11
; 4b. ๐ฅ = โ
8
63
4c. ๐๐ ๐๐๐ฟ๐๐๐ผ๐๐ ; 4d. ๐ฅ = 7 ;
4e. ๐ผ๐๐น๐ผ๐๐ผ๐๐ธ๐ฟ๐ ๐๐ด๐๐ ๐๐๐ฟ๐๐๐ผ๐๐๐, โ โ {
1
2
4f. ๐๐ ๐๐๐ฟ๐๐๐ผ๐๐ ;
