1. 9.6 Solving Rational Equations
4/30/2014

2. Vocabulary 2 3 = 6 9 If 𝑎 𝑏 = 𝑐 𝑑 Then a∙𝑑=𝑏∙𝑐 Then 2∙9=3∙6 18 = 18
Rational Equation:
Equation that shows two rational expressions or fractions are equal.
2 3 = 6 9
Then 2∙9=3∙6
18 = 18
Example:
If 𝑎 𝑏 = 𝑐 𝑑
Then a∙𝑑=𝑏∙𝑐
Cross Multiply:

3. Example 1
Solve for x: = 𝑥 7
3∙7=4∙𝑥
21=4𝑥 4
21 4 =𝑥

4. Example 2 2(𝑥+3)=3∙𝑥 2𝑥+6=3𝑥 −2𝑥 −2𝑥 6=𝑥 2 3 = 6 9 2∙9=3∙6 18 = 18
Solve for x: = 𝑥 𝑥+3
To check: = 6 6+3
2 3 = 6 9
2∙9=3∙6
18 = 18
2(𝑥+3)=3∙𝑥
2𝑥+6=3𝑥
−2𝑥 −2𝑥
6=𝑥

5. Example 3 2 𝑥−2 =𝑥 𝑥−2 2𝑥−4= 𝑥 2 −2𝑥 −2𝑥+4 −2𝑥+4 0= 𝑥 2 −4𝑥+4
Solve for x: 2 𝑥−2 = 𝑥 𝑥−2
Or if you just look at the problem you can easily see that for the 2 fractions to be equal x must be 2!
2 𝑥−2 =𝑥 𝑥−2
2𝑥−4= 𝑥 2 −2𝑥
−2𝑥 −2𝑥+4
0= 𝑥 2 −4𝑥+4
0=(𝑥−2)(𝑥−2)
2=𝑥
2 is called an Extraneous solution because it leads to a division by 0 in the original equation.
Always check for extraneous solutions!
If x = 2, what happens to the denominator?
Yes, it becomes 0 and you cannot divide by 0.
So there is NO SOLUTION!

6. Example 4 𝑥 𝑥−1 =2 𝑥+5 𝑥 2 −𝑥 =2𝑥+10 −2𝑥−10 −2𝑥−10 𝑥 2 −3𝑥−10=0
Solve for x: 𝑥 𝑥+5 = 2 𝑥−1
To check: = 2 5−1
5 10 = 2 4
20=20
𝑥 𝑥−1 =2 𝑥+5
𝑥 2 −𝑥 =2𝑥+10
−2𝑥−10 −2𝑥−10
𝑥 2 −3𝑥−10=0
(𝑥−5)(𝑥+2) =0
𝑥= 5,−2
To check: −2 −2+5 = 2 −2−1
−2 3 = 2 −3

7. Now we’re going to solve rational equations where one side contains addition or subtraction. Ex. 2 𝑥 𝑥 =2 How? By multiplying each term by the least common denominator (LCD). That will get rid of all fractions!!!

8. Example 5 2 𝑥 2 + 3 𝑥 =2 2+3𝑥= 2𝑥 2 −2−3𝑥 −3𝑥−2 0 = 2𝑥 2 −3𝑥−2
Solve for x: 𝑥 𝑥 =2
What’s the LCD?
LCD: 𝑥 2
2 𝑥 𝑥 =2
2+3𝑥= 2𝑥 2
−2−3𝑥 −3𝑥−2
0 = 2𝑥 2 −3𝑥−2
0=(𝑥−2)(2𝑥+1)
− 1 2 , 2=𝑥
•𝑥 2
•𝑥 2
•𝑥 2 1
2(-2) = 2 2 -4 1 -2 -3

9. Example 6 Solve for x: 8+ 2 𝑥−1 = 2𝑥 𝑥−1
8+ 2 𝑥−1 = 2𝑥 𝑥−1 8(𝑥−1)+ 2 𝑥−1 (𝑥−1)= 2𝑥 𝑥−1 (𝑥−1) 8𝑥−8+2=2𝑥 8𝑥−6=2𝑥 −2𝑥+6 −2𝑥+6 6𝑥=6 𝑥=1
LCD: 𝑥−1
Multiply each term by LCD
Distributive Prop
No Solution

10. Example 7 Solve for x: 2 𝑥−1 + 3 𝑥 =2
LCD: 𝑥(𝑥−1)
Multiply each term by LCD
Distributive Prop
Factor using Big X

11. Homework:
9.6 p.503 #21-41 odd only
Since you can check your answers in the back of the book (and you should), you must show work!!!
“I’m glad I know sign language, it’s pretty handy”