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Lesson Menu Five-Minute Check (over Lesson 9–5) Then/Now New Vocabulary Example 1:Solve a Rational Equation Example 2:Solve a Rational Equation Example 3:Real-World Example: Mixture Problem Example 4:Real-World Example: Distance Problem Example 5:Real-World Example: Work Problems Key Concept: Solving Rational Inequalities Example 6:Solve a Rational Inequality

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 1 A.direct; B.joint; C.inverse; 2 D.combined; 2 State whether represents a direct, joint, inverse, or combined variation. Then name the constant of variation.

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 2 A.direct; 7.5 B.joint; 7.5 C.inverse; D.combined; 7.5 State whether 7.5x = y represents a direct, joint, inverse, or combined variation. Then name the constant of variation.

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 3 A.4.8 B.6.4 C.8.6 D.10.2 If y varies inversely as x and y = 8 when x = 12, find y when x = 15.

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 4 A.9 B.11 C.13 D.15 If y varies jointly as x and z and y = 45 when x = 10 and z = 3, find y when x = 2 and z = 5.

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 5 A mi B mi C.325 mi D.260 mi A map shows the scale 1.5 inches equals 65 miles. How many miles apart are two cities if they are 7.5 inches apart on the map?

Over Lesson 9–5 A.A B.B C.C D.D 5-Minute Check 6 A.2% B.3.5% C.4% D.4.5% The amount of interest earned on a savings account varies jointly with time and the amount deposited. After 5 years, interest on $1000 in the savings account is $225. What is the annual interest rate (constant of variation)?

Then/Now You simplified rational expressions. (Lesson 9–2) Solve rational equations. Solve rational inequalities.

Vocabulary rational equation weighted average rational inequality

Example 1 Solve a Rational Equation The LCD for the terms is 24(3 – x). Original equation Solve. Check your solution. Multiply each side by 24(3 – x).

Example 1 Solve a Rational Equation Distributive Property Simplify. Add 6x and –63 to each side.

Example 1 Solve a Rational Equation Original equation Check x = –45 Simplify. The solution is correct.

Example 1 Solve a Rational Equation Answer:The solution is –45.

A.A B.B C.C D.D Example 1 Solve. A.–2 B. C. D.2

Example 2 Solve a Rational Equation The LCD is (p + 1)(p – 1). Original equation Solve Check your solution. Multiply by the LCD.

Example 2 Solve a Rational Equation (p – 1)(p 2 – p – 5) = (p 2 – 7)(p + 1) + p(p + 1)(p – 1) p 3 – p 2 – 5p – p 2 + p + 5 = p 3 + p 2 – 7p – 7 + p 3 – p p 3 – 2p 2 – 4p + 5 = 2p 3 + p 2 – 8p – 7 0= p 3 + 3p 2 – 4p – 12 Divide common factors. Distributive Property Simplify. Subtract p 3 – 2p 2 – 4p + 5 from each side.

Example 2 Solve a Rational Equation Zero Product Property 0=(p + 3)(p + 2)(p – 2) Factor. 0=p + 3 or 0 = p + 2 or 0 = p – 2 Original equation Check Try p = –3. p = –3

Example 2 Solve a Rational Equation Simplify. Original equation Try p = –2.

Example 2 Solve a Rational Equation Simplify. p = –2 Simplify.

Example 2 Solve a Rational Equation Answer: The solutions are –3, –2 and 2. Try p = 2. Original equation Simplify. p = 2 Simplify.

A.A B.B C.C D.D Example 2 A.4 B.–2 C.2 D.–4

Example 3 Mixture Problem BRINE Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine? UnderstandAaron needs to know how much of a solution needs to be added to an original solution to create a new solution.

Example 3 Mixture Problem PlanEach solution has a certain percentage that is salt. The percentage of brine in the final solution must equal the amount of brine divided by the total solution. Percentage of brine in solution

Example 3 Mixture Problem Substitute. Simplify numerator. LCD is 100(10 + x). SolveWrite a proportion.

Example 3 Mixture Problem Distribute. Subtract 50x and 160. Divide each side by 30. Answer: Aaron needs to add ounces of 80% brine solution. Simplify. Divide common factors.

Example 3 Mixture Problem Simplify. CheckOriginal equation ? ? 0.5 = 0.5 Simplify.

A.A B.B C.C D.D Example 3 A.9.6 ounces B.10.4 ounces C.11.8 ounces D.12.3 ounces BRINE Janna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base?

Example 4 Distance Problem SWIMMING Lilia swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water? UnderstandWe are given the speed of the current, the distance she swims upstream, and the total time. PlanShe swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is d = rt or

Example 4 Distance Problem Solve Original equation Time going with the currentplus time going against the currentequals total time. 5 Let r equal her speed in still water. Then her speed with the current is r + 1, and her speed against the current is r – 1.

Example 4 Distance Problem Divide Common Factors Distribute. Simplify. Subtract 4r from each side. (r + 1)2 + (r – 1)2 = 5(r 2 – 1) Simplify. Multiply each side by r 2 – 1.

Example 4 Distance Problem Use the Quadratic Formula to solve for r. Quadratic Formula x = r, a = 5, b = – 4, and c = –5 Simplify.

Example 4 Distance Problem Answer: Since speed must be positive, the answer is about 1.5 miles per hour. Check Original equation r ≈ 1.5 or –0.7Use a calculator. r = 1.5 ? Simplify. Simplify. ?

A.A B.B C.C D.D Example 4 A.about 0.6 mph B.about 2.0 mph C.about 4.6 mph D.about 6.6 mph SWIMMING Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water?

Example 5 Work Problems MOWING LAWNS Wuyi and Uima mow lawns together. Wuyi working alone could complete a particular job in 4.5 hours, and Uima could complete it alone in 3.7 hours. How long does it take to complete the job when they work together? UnderstandWe are given how long it takes Wuyi and Uima working alone to mow a particular lawn. We need to determine how long it would take them together. PlanWuyi can mow the lawn in 4.5 hours, so the rate of mowing is of a lawn per hour.

Example 5 Work Problems Uima can mow the lawn in 3.7 hours, so the rate of mowing is of a lawn per hour. The combined rate is

Example 5 Work Problems Multiply both sides by x. Solve Write the equation. Add Answer: It would take Wuyi and Uima about 2 hours to mow the lawn together. x ≈ Multiply 1 by

Example 5 Work Problems x ≈ 2 Simplify. Check Original equation ?

A.A B.B C.C D.D Example 5 A.about 2 hours and 28 minutes B.about 2 hours and 36 minutes C.about 2 hours and 45 minutes D.about 2 hours and 56 minutes PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular job in 6.4 hours, and Monique could complete it alone in 4.8 hours. How long does it take to complete the job when they work together?

Concept

Example 6 Solve a Rational Inequality Step 1Values that make the denominator equal to 0 are excluded from the denominator. For this inequality the excluded value is 0. Related equation Step 2Solve the related equation. Solve

Example 6 Solve a Rational Inequality Multiply each side by 9k. Simplify. Add. Divide each side by 6.

Example 6 Solve a Rational Inequality Step 3Draw vertical lines at the excluded value and at the solution to separate the number line into regions. Now test a sample value in each region to determine if the values in the region satisfy the inequality.

Example 6 Solve a Rational Inequality Test k = –1. k < 0 is a solution.

Example 6 Solve a Rational Inequality Test k =. 0 < k < is not a solution. 

Example 6 Solve a Rational Inequality Test k = 1.

A.A B.B C.C D.D Example 6 A.x < 0 B.x > 0 C.x 4 D.0 < x < 4 Solve.

End of the Lesson