1. 10-7 Solving Rational Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 10-7 Solving Rational Equations Warm Up 1. Find the LCM of x, 2x 2, and 6. 2. Find the LCM of p 2 – 4p and p 2 – 16. Multiply. Simplify your answer. 3. 4. 5.

3. 10-7 Solving Rational Equations Preparation for 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. California Standards

4. 10-7 Solving Rational Equations rational equation extraneous solutions Vocabulary

5. 10-7 Solving Rational Equations A rational equation is an equation that contains one or more rational expressions. If a rational equation is a proportion, it can be solved using the Cross Product Property.

6. 10-7 Solving Rational Equations Additional Example 1: Solving Rational Equations by Using Cross Products Use cross products. 5x = (x – 2)(3) 5x = 3x – 6 2x = – 6 Solve. Check your answer. x = – 3 Distribute 3 on the right side. Subtract 3x from both sides. Divide both sides by 2.

7. 10-7 Solving Rational Equations Additional Example 1 Continued Check – 1 – 1 Substitute –3 for x in the original equation.

8. 10-7 Solving Rational Equations Check It Out! Example 1a Solve. Check your answer. Use cross products. Distribute 1 on the right side. Subtract n from both sides. 3n = (n + 4)(1) 3n = n + 4 2n = 4 n = 2 Divide both sides by 2.

9. 10-7 Solving Rational Equations Check It Out! Example 1a Continued Check Substitute 2 for n in the original equation.

10. 10-7 Solving Rational Equations Check It Out! Example 1b Solve. Check your answer. 4h = (h + 1)(2) 4h = 2h + 2 2h = 2 h = 1 Use cross products. Distribute 2 on the right side. Subtract 2h from both sides. Divide both sides by 2.

11. 10-7 Solving Rational Equations Check It Out! Example 1b Continued Check Substitute 1 for h in the original equation.

12. 10-7 Solving Rational Equations Check It Out! Example 1c Solve. Check your answer. 21x = (x – 7)(3) 21x = 3x – 21 18x = – 21 x = Use cross products. Distribute 3 on the right side. Subtract 3x from both sides. Divide both sides by 18.

13. 10-7 Solving Rational Equations Check Check It Out! Example 1c Continued Substitute for x in the original equation.

14. 10-7 Solving Rational Equations Some rational equations contain sums or differences of rational expressions. To solve these, you must find a common denominator for all the rational expressions in the equation.

15. 10-7 Solving Rational Equations Additional Example 2A: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD. 2x(x + 1) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

16. 10-7 Solving Rational Equations Additional Example 2A Continued Step 3 Simplify and solve. (2x)(2) + 6(x + 1) = 5(x + 1) 4x + 6x + 6 = 5x + 5 10x + 6 = 5x + 5 5x = – 1 Divide out common factors. Simplify. Distribute and multiply. Combine like terms. Subtract 5x and 6 from both sides. Divide both sides by 5.

17. 10-7 Solving Rational Equations Additional Example 2A Continued Check

18. 10-7 Solving Rational Equations Additional Example 2B: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD. (x2)(x2) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

19. 10-7 Solving Rational Equations Additional Example 2B Continued Step 3 Simplify and solve. Divide out common factors. 4x – 3 = x 2 – x 2 + 4x – 3 = 0 x 2 – 4x + 3 = 0 (x – 3)(x – 1) = 0 x = 3 or 1 Simplify. Subtract x 2 from both sides. Factor. Solve. Multiply by – 1.

20. 10-7 Solving Rational Equations Additional Example 2B Continued Check

21. 10-7 Solving Rational Equations Check It Out! Example 2a Solve each equation. Check your answer. Step 1 Find the LCD. a(a +1) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

22. 10-7 Solving Rational Equations Check It Out! Example 2a Continued Step 3 Simplify and solve. Divide out common factors. 3a = 4(a + 1) 3a = 4a + 4 – 4 = a Simplify. Distribute the 4. Subtract the 4 and 3a from both sides.

23. 10-7 Solving Rational Equations Check It Out! Example 2a Continued Check

24. 10-7 Solving Rational Equations Check It Out! Example 2b Solve each equation. Check your answer. Step 1 Find the LCD. 2j(j +2) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side.

25. 10-7 Solving Rational Equations Check It Out! Example 2b Continued Solve each equation. Check your answer. 12j – 10(2j + 4) = 4j + 8 12j – 20j – 40 = 4j + 8 – 12j = 48 j = – 4 Simplify. Distribute 10. Combine like terms. Divide out common factors. Divide both sides by –12.

26. 10-7 Solving Rational Equations Check It Out! Example 2b Continued Check

27. 10-7 Solving Rational Equations When you multiply each side of an equation by the LCD, you may get an extraneous solution. An extraneous solution is a solution to a resulting equation that is not a solution to the original equation. Because of extraneous solutions, it is especially important to check your answers.

28. 10-7 Solving Rational Equations Additional Example 3: Extraneous Solutions Solve. Check your answer. 2(x 2 – 1) = (x + 1)(x – 6) 2x 2 – 2 = x 2 – 5x – 6 x 2 + 5x + 4 = 0 (x + 1)(x + 4) = 0 x = – 1 or x = – 4 Use cross products. Distribute 2 on the left side. Multiply the right side. Subtract x 2 from both sides. Add 5x and 6 to both sides. Factor the quadratic expression. Use the Zero Product Property. Solve.

29. 10-7 Solving Rational Equations Additional Example 3 Continued Solve. Check your answer.  Because and are undefined, –1 is not a solution. – 1 is an extraneous solution. The only solution is – 4. Check

30. 10-7 Solving Rational Equations Check It Out! Example 3a Solve. Check your answer. (x – 2)(x – 7) = 3(x – 7) Use cross products. Distribute 3 on the right side. Multiply the left side. x 2 – 9x + 14 = 3x – 21 x 2 – 12x + 35 = 0 Subtract 3x from both sides. Add 21 to both sides. (x – 7)(x – 5) = 0 x = 7 or x = 5 Factor the quadratic expression. Use the Zero Product Property. Solve.

31. 10-7 Solving Rational Equations Check It Out! Example 3a Continued 7 is an extraneous solution. The only solution is 5. Because and are undefined, 7 is not a solution.  Check

32. 10-7 Solving Rational Equations Check It Out! Example 3b Solve. Check your answer. (x + 1)(x – 3) = 4(x – 2) Use cross products. Distribute 4 on the right side. Multiply the left side. x 2 – 2x – 3 = 4x – 8 x 2 – 6x + 5 = 0 Subtract 4x from both sides. Add 8 to both sides. (x – 1)(x – 5) = 0 x = 1 or x = 5 Factor the quadratic expression. Use the Zero Product Property. Solve.

33. 10-7 Solving Rational Equations Check It Out! Example 3b Continued There are no extraneous solutions. The solutions are 1 and 5. 1 and 5 are solutions. Check

34. 10-7 Solving Rational Equations Check It Out! Example 3c Solve. Identify any extraneous solutions. 6(x 2 + 2x) = 9(x 2 ) Use cross products. Distribute 6 on the left side. Multiply the right side. 3x 2 – 12x = 0 3x(x – 4) = 0 3x = 0, or x – 4 = 0 x = 0 or x = 4 Factor the quadratic expression. Use the Zero Product Property. Solve. Subtract 9x 2 from both sides. Multiply through with – 1. 6x 2 + 12x) = 9x 2

35. 10-7 Solving Rational Equations Check It Out! Example 3c Continued Check 0 is an extraneous solution. The only solution is 4. Because and are undefined, 0 is not a solution. 

36. 10-7 Solving Rational Equations Lesson Quiz Solve each equation. Check your answer. 3. 2. 1.24 – 4, 3 4. Solve. Check your answer. –5–5