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Section 9.6

Section 9.6. Rational Equations. Solving Rational Equations. ALGEBRA 2 LESSON 9-6. (For help, go to Lessons 5-8 and 9-1.). Find the LCD of each pair of fractions. 1. , 2. , 3. , 4. , 5. , 6. , . x 2. 3 x 8. 1 3 t. 1 5 t 2. 2 h h 3. 4 y + 2. 3

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Section 9.6

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  1. Section 9.6 Rational Equations

  2. Solving Rational Equations ALGEBRA 2 LESSON 9-6 (For help, go to Lessons 5-8 and 9-1.) Find the LCD of each pair of fractions. 1. , 2. , 3. , 4. , 5. , 6. , x 2 3x 8 1 3t 1 5t2 2h h3 4 y + 2 3 y – 1 4 3h2 1 k + 2 3k k2 – 4 z 2z + 1 1 z Check Skills You’ll Need 9-6

  3. Solving Rational Equations ALGEBRA 2 LESSON 9-6 Solutions 1. 3t = 3 • t; 5t2 = 5 • t • t; LCD = 3 • 5 • t • t = 15t2 2. 2 = 2; 8 = 2 • 2 • 2; LCD = 2 • 2 • 2 = 8 3. 3h2 = 3 • h • h; h3 = h • h • h; LCD = 3 • h • h • h = 3h3 4.y + 2 = y + 2; y – 1 = y – 1;LCD = (y + 2)(y – 1) or y2 + y – 2 5. 2z + 1 = 2z + 1; z = z; LCD = z(2z + 1) or 2z2 + z 6.k + 2 = k + 2; k2 – 4 = (k + 2)(k – 2); LCD = (k + 2)(k – 2) or k2 – 4 9-6

  4. 1 x – 3 6x x 2– 9 = 5x – 3 = 0 or x – 3 = 0 x = or x = 3 Zero-Product Property 3 5 Solving Rational Equations ALGEBRA 2 LESSON 9-6 1 x – 3 6x x 2 – 9 Solve = . x2 – 9 = 6x(x – 3) Write the cross products. x2 – 9 = 6x2 – 18xDistributive Property –5x2 + 18x – 9 = 0 Write in standard form. 5x2 – 18x + 9 = 0 Multiply each side by –1. (5x – 3)(x – 3) = 0 Factor. 9-6

  5. Solving Rational Equations ALGEBRA 2 LESSON 9-6 (continued) Check: When x = 3, both denominators in the original equation are zero. The original equation is undefined at x = 3. So x = 3 is not a solution. When is substituted for x in the original equation, both sides equal – . 35 5 12 Quick Check 9-6

  6. 3 5x 4 3x 1 3 – = . 1 3 3 5x 4 3x 15x – = 15xMultiply each side by the LCD, 15x. 15x 3 45x 5x 60x 3x – = Distributive Property 11 5 – = x 11 5 11 5 Since – makes the original equation true, the solution is x = – . Solving Rational Equations ALGEBRA 2 LESSON 9-6 3 5x 4 3x 1 3 Solve – = . 9 – 20 = 5xSimplify. Quick Check 9-6

  7. Define: Distance (mi) Rate (mi/h) Time (h) With current 6 2 + r Against current 4 2 – r 6 (2 + r) 4 (2 – r) 6 (2 + r) 4 (2 – r) Write: = Solving Rational Equations ALGEBRA 2 LESSON 9-6 Josefina can row 4 miles upstream in a river in the same time it takes her to row 6 miles downstream. Her rate of rowing in still water is 2 miles per hour. Find the speed of the river current. Relate: speed with the current = speed in still water + speed of the current, speed against the current = speed in still water – speed of the current, time to row 4 miles upstream = time to row 6 miles downstream 9-6

  8. 6 (2 + r) 4 (2 – r) = 4 (2 – r) 6 (2 + r) (2 + r )(2 – r ) = (2 + r )(2 – r ) Multiply by the LCD (2 + r )(2 – r ). Solving Rational Equations ALGEBRA 2 LESSON 9-6 (continued) (2 – r )(6) = (2 + r )(4) Simplify. 12 – 6r = 8 + 4rDistributive Property 4 = 10rSolve for r. 0.4 = rSimplify. The speed of the river current is 0.4 mi/h. Quick Check 9-6

  9. Relate: Jim’s work speed + Alberto’s work speed = combined work speed Define: Time (hours) Rate (square feet per hour) Jim 2x Alberto x Combined 15 6000 2x 6000 x 6000 15 = 400 6000 2x 6000 x Write:+ = 400 Solving Rational Equations ALGEBRA 2 LESSON 9-6 Jim and Alberto have to paint 6000 square feet of hallway in an office building. Alberto works twice as fast as Jim. Working together, they can complete the job in 15 hours. How long would it take each of them working alone? 9-6

  10. 6000 2x 6000 x + = 400 6000 2x 6000 x 2x + = 2x(400) Multiply by the LCD, 2x. 2x(6000) 2x 2x(6000) x + = 2x(400) Distributive Property Solving Rational Equations ALGEBRA 2 LESSON 9-6 (continued) 6000 + 12000 = 800xSimplify. 18000 = 800xSimplify. 22.5 = xSolve for x. Alberto could paint the hallway in 22.5 hours. Jim could paint the hallway in 2(22.5) hours or 45 hours. Quick Check 9-6

  11. Solving Rational Equations ALGEBRA 2 LESSON 9-6 Solve each equation. Check each solution. 1. = 2. 1 + = 3. + x = 4. The speed of the current in a river is 5 miles per hour. A boat leaves a dock on the bank of the river, travels upstream 25 miles, and returns to the dock in 12 hours. What is the speed of the boat in still water? 2 x– 5 10 3 2 x 3 x2 5x x2 – 25 –3, 1 4x – 3 x– 1 x x– 1 3 7.5 mi/h 9-6

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