9.6 Solving Rational Equations 4/30/2014

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:

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

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: 2 3 = 𝑥 𝑥+3 To check: 2 3 = 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

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+4 −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!

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: 5 5+5 = 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

Now we’re going to solve rational equations where one side contains addition or subtraction. Ex. 2 𝑥 2 + 3 𝑥 =2 How? By multiplying each term by the least common denominator (LCD). That will get rid of all fractions!!!

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

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

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

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”