WARM UP Divide and check

SOLVING RATIONAL EXPRESSIONS

USING RATIONAL EXPRESSIONS A rational expression is an equation that contains one or more rational expressions. These are rational expressions: To solve a rational expression, we multiply both sides by the LCD to clear fractions.

EXAMPLE 1 Solve When clearing fractions, be sure to multiply every term in the equation by the LCD. This yields an equation without rational expressions, which can be solved directly. We must first find the LCD: 6x Check:

TRY THIS… Solve a. b.

EXAMPLE 2 When we multiply both sides of an equation by an expression containing a variable, we may not get an equivalent equation. 5 is not a solution of the original equation because it results in division by 0. Since 5 is the only possible solution, the equation has no solution. The new equation may have solutions that the original one does not. Thus we must always check possible solutions in the original equation. Solve: The LCD is is x – 5. We multiply by x – 5 to clear fractions. Check by substituting 5 for x :

TRY THIS… Solve a. b.

EXAMPLE 3 The number 2 is a solution but -2 is not since it results in division by 0. Solve: The LCD is is x – 2. We multiply by x – 2. Check by substituting -2 for x : or Check by substituting 2 for x :

EXAMPLE 4 The solutions are 2 and 3. Solve: The LCD is is x. We multiply by x. Check by substituting 3 for x : or Check by substituting 2 for x :

TRY THIS… Solve a. b. c. d.

EXAMPLE 5 This checks in the original equation, so the solution is 7 Solve: The LCD is is (x + 5) and (x – 5). We multiply both sides of the equation by (x + 5) and (x – 5). 1

TRY THIS… Solve a. b.

WORK PROBLEMS  Tom knows that he can mow a golf course in 4 hours. He also knows that Perry takes 5 hours to mow the same course. Tom must complete the job in 2-1/2 hours. Can he and Perry get the job done in time? How long will it take them to complete the job together? Solving Work Problems If a job can be done in t hours, then of it can be done in one hour. (The above condition holds for any unit of time.)

SOLVING THE PROBLEM  UNDERSTAND the problem. Question: How long will it take the two of them to mow the lawn together? Data: Tom takes 4 hours to mow the lawn. Perry takes 5 hours to mow the lawn.  Develop and carry out a PLAN. Let t represent the total number of hours it takes them working together. Then they can mow of it in 1 hour.

TRANSLATION We can now translate to an equation. Translating to an equation We solve the equation. Multiplying on both sides by the LCD to clear fraction. hours

PROBLEM CONTINUED  Find the ANSWER and CHECK. Tom can do the entire job in 4 hours, so he can do just over half in 2 hours. Perry can do the entire job in 5 hours, so he can do just under half the job in 2 hours. It is reasonable that working together they can do the job in 2 hours. It will take them 2 hours together, so they will finish in time.

TRY THIS… Carlos can do a typing job in 6 hours. Lynn can do the same job in 4 hours. How long would it take them to do the job working together with two typewriters.

CH. 6.6/6.7 HOMEWORK  Textbook pg. 269 #2, 6, 10, 14 & 22  pg. 273 # 2, 4, 6 & 8