5-Minute Check on Chapter 2 Transparency 3-1 Click the mouse button or press the Space Bar to display the answers. 1.Evaluate 42 - |x - 7| if x = -3 2.Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) 5.A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green? 6. Which of the following is a true statement Standardized Test Practice: ACBD 8/4 < 4/8-4/8 < -8/4-4/8 > -8/4-4/8 > 4/8

Lesson 12-9 Solving Rational Equations

Transparency 9 Click the mouse button or press the Space Bar to display the answers.

Transparency 9a

Objectives Solve rational equations Eliminate extraneous solutions

Vocabulary xxxxxx –

Key Concept Step 1: Explore the Problem –Identify what information is given (the facts) –Identify what you are asked to find (the question) Step 2: Plan the Solution –Find an equation the represents the problem –Let a variable represent what you are looking for Step 3: Solve the Problem –Plug into your equation and solve for the variable Step 4: Examine the Solution –Does your answer make sense? –Does it fit the facts in the problem?

Example 1 Solve Original equation Cross multiply. Distributive Property Add –2x and 48 to each side. Answer: Divide each side by 6.

Example 2 Solve Original equation The LCD is Distributive Property

Example 2 cont Simplify. Add. Subtract 1 from each side. Divide each side by 6. Answer:

Example 3 Solve Distributive Property Original equation The LCD is

Example 3 cont Simplify. or Set equal to 0. Factor. Answer:The number - 1 is an excluded value for x. Thus, the solution is 3. CheckCheck solutions by substituting each value in the original equation.

Example 4 TV Installation On Saturdays, Lee helps her father install satellite TV systems. The jobs normally take Lee’s father about 2½ hours. But when Lee helps, the jobs only take them 1½ hours. If Lee were installing a satellite system herself, how long would the job take? ExploreSince it takes Lee’s father 2½ or 5/2 hours to install one job, he can finish 2/5 of the job in one hour. The amount of work Lee can do in one hour can be represented by 1/t. To determine how long it takes Lee to do the job, use the formula: Lee’s work + her father’s work = 1 job.

Example 4 cont PlanThe time that both of them worked was 1½ hours. Each rate multiplied by this time results in the amount of work done by each person. SolveLee’sher father’stotal workplus workequalswork. 1 Multiply. The LCD is 10t.

Example 4 cont Distributive Property Simplify. Add –6t to each side. Divide each side by 4. Answer:The job would take Lee or hours by herself. ExamineThis seems reasonable because the combined efforts of both took longer than half of her father’s usual time.

Example 5 Transportation The schedule for the Washington, D.C., Metrorail is shown to the right. Suppose two Red Line trains leave their stations at opposite ends of the line at exactly 2:00 P.M. One train travels between the two stations in 48 minutes and the other train takes 54 minutes. At what time do the two trains pass each other? Determine the rates of both trains. The total distance is 19.4 miles. Train 1 Train 2

Example 5 cont Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 19.4 miles. t min Train 2 t min Train 1 d = r  t tr The sum of the distances is 19.4.

Example 5 cont The LCD is 432. Distributive Property Simplify. Add. Divide each side by Answer:The trains passed each other at about 25 minutes after they left their stations, at 2:25 P.M.

Example 6 Solve Original equation The LCD is x – 1. Distributive Property Simplify. Subtract 2 from each side Answer:Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus, the equation has no solution.

Example 7 Solve Original equation The LCD is x – Distributive Property

Example 7 cont Simplify. Subtract 4 from each side. Factor. or Zero Product Property Answer:The number 2 is an extraneous solution, since 2 is an excluded value for x. Thus, –2 is the solution of the equation.

Summary & Homework Summary: –Use cross products to solve rational equations with a single fraction on each side of the equal sign –Multiply every term of a more complicated rational equation by the LCD to eliminate fractions Homework: –none