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12-7Solving 12-7 SolvingRational RationalEquations Equations Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 11 12-7 Solving Rational Equations Warm Up 1. Find the LCM of x, 2x2, and 6. 2. Find the LCM of p2 – 4p and p2 – 16. Multiply. Simplify your answer. 3. 5. Holt Algebra 1 4. 12-7 Solving Rational Equations Objectives Solve rational equations. Identify extraneous solutions. Holt Algebra 1 12-7 Solving Rational Equations Vocabulary rational equation Holt Algebra 1 12-7 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. Holt Algebra 1 12-7 Solving Rational Equations Example 1: Solving Rational Equations by Using Cross Products Solve . Check your answer. Use cross products. 5x = (x – 2)(3) 5x = 3x – 6 2x = –6 x = –3 Check Distribute 3 on the right side. Subtract 3x from both sides. –1 –1 Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 1a Solve . Check your answer. Check Use cross products. 3n = (n + 4)(1) 3n = n + 4 2n = 4 n=2 Holt Algebra 1 Distribute 1 on the right side. Subtract n from both sides. Divide both sides by 2. 12-7 Solving Rational Equations Check It Out! Example 1b Solve . Check your answer. Check Use cross products. 4h = (h + 1)(2) 4h = 2h + 2 Distribute 2 on the right side. 2h = 2 Subtract 2h from both sides. h=1 Divide both sides by 2. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 1c Solve . Check your answer. Check Use cross products. 21x = (x – 7)(3) 21x = 3x –21 18x = –21 x= Holt Algebra 1 Distribute 3 on the right side. Subtract 3x from both sides. Divide both sides by 18. 12-7 Solving Rational Equations Some rational equations contain sums or differences of rational expressions. To solve these, you must find the LCD of all the rational expressions in the equation. Holt Algebra 1 12-7 Solving Rational Equations 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. Holt Algebra 1 12-7 Solving Rational Equations Example 2A Continued Step 3 Simplify and solve. Divide out common factors. (2x)(2) +6(x +1) = 5(x +1) 4x + 6x + 6 = 5x + 5 10x + 6 = 5x + 5 5x = –1 Holt Algebra 1 Simplify. Distribute and multiply. Combine like terms. Subtract 5x and 6 from both sides. Divide both sides by 5. 12-7 Solving Rational Equations Example 2A Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Example 2B: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD (x2) Include every factor of the denominator. Step 2 Multiply both sides by the LCD Distribute on the left side. Holt Algebra 1 12-7 Solving Rational Equations Example 2B Continued Step 3 Simplify and solve. Divide out common factors. 4x – 3 = x2 –x2 + 4x – 3 = 0 x2 – 4x + 3 = 0 (x – 3)(x – 1) = 0 Simplify. Subtract x2 from both sides. Multiply by – 1. Factor. x = 3 or 1 Solve. Holt Algebra 1 12-7 Solving Rational Equations Example 2B Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 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. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 2a Continued Step 3 Simplify and solve. Divide out common factors. 3a = 4(a + 1) Simplify. 3a = 4a + 4 –4 = a Holt Algebra 1 Distribute the 4. Subtract the 4 and 3a from both sides. 12-7 Solving Rational Equations Check It Out! Example 2a Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 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. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 2b Continued Solve each equation. Check your answer. Divide out common terms. 12j – 10(2j + 4) = 4j + 8 12j – 20j – 40 = 4j + 8 –12j = 48 j = –4 Holt Algebra 1 Simplify. Distribute 10. Combine like terms. 12-7 Solving Rational Equations Check It Out! Example 2b Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 2c Solve each equation. Check your answer. Step 1 Find the LCD t(t +3) Include every factor of the denominator. Step 2 Multiply both sides by the LCD Distribute on the right side. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 2c Continued Solve each equation. Check your answer. Divide out common terms. 8t = (t + 3) + t(t + 3) 8t = t + 3 + t2 + 3t 0 = t2 – 4t + 3 Distribute t. Combine like terms. 0 = (t – 3)(t – 1) Factor. t = 3, 1 Holt Algebra 1 Simplify. 12-7 Solving Rational Equations Check It Out! Example 2c Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Example 3: Problem-Solving Application Copy machine A can make 200 copies in 60 minutes. Copy machine B can make 200 copies in 10 minutes. How long will it take both machines working together to make 200 copies? Holt Algebra 1 12-7 Solving Rational Equations 1 Understand the Problem The answer will be the number of minutes m machine A and machine B need to print the copies. List the important information: • Machine A can print the copies in 60 minutes, which is of the job in 1 minute. • Machine B can print the copies in 10 minutes, which is of the job in 1 minute. Holt Algebra 1 12-7 Solving Rational Equations 2 Make a Plan The part of the copies that machine A can print plus the part that machine B can print equals the complete job. Machine A’s rate times the number of minutes plus machine B’s rate times the number of minutes will give the complete time to print the copies. (machine A’s rate) m Holt Algebra 1 m + (machine B’s rate) + m m = complete job = 1 12-7 Solving Rational Equations 3 Solve Multiply both sides by the LCD, 60. 1m + 6m = 60 7m = 60 Distribute 60 on the left side. Combine like terms. Divide both sides by 7. Machine A and Machine B working together can print the copies in a little more than 8.5 minutes. Holt Algebra 1 12-7 Solving Rational Equations 4 Look Back Machine A prints of the copies per minute and machine B prints of the copies per minute. So in minutes, machine A prints of the copies and machine B prints of the copies. Together, they print Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 3 Cindy mows a lawn in 50 minutes. It takes Sara 40 minutes to mow the same lawn. How long will it take them to mow the lawn if they work together? 1 Understand the Problem The answer will be the number of minutes m Sara and Cindy need to mow the lawn. Holt Algebra 1 12-7 Solving Rational Equations 1 Understand the Problem The answer will be the number of minutes m Sara and Cindy need to mow the lawn. List the important information: • Cindy can mow the lawn in 50 minutes, which is of the job in 1 minute. • Sara can mow the lawn in 40 minutes, which is of the job in 1 minute. Holt Algebra 1 12-7 Solving Rational Equations 2 Make a Plan The part of the lawn Cindy can mow plus the part of the lawn Sara can mow equals the complete job. Cindy’s rate times the number of minutes plus Sara’s rate times the number of minutes will give the complete time to mow the lawn. (Cindy’s rate) m + (Sara’s rate) m = lawn mowed m Holt Algebra 1 + m = 1 12-7 Solving Rational Equations 3 Solve Multiply both sides by the LCD, 200. 5m + 4m = 200 9m = 200 Distribute 200 on the left side. Combine like terms. Divide both sides by 9. Cindy and Sara working together can mow the lawn in a little more than 22.2 minutes. Holt Algebra 1 12-7 Solving Rational Equations 4 Look Back Cindy mows mows of the lawn in 1 minute and Sara of the lawn in 1 minute. So, in minutes Cindy mows Sara mows mow Holt Algebra 1 of the lawn and of the lawn. Together they lawn. 12-7 Solving Rational Equations When you multiply each side of an equation by the LCD, you may get an extraneous solution. Recall from Chapter 11 that an extraneous solution is a solution to a resulting equation that is not a solution to the original equation. Holt Algebra 1 12-7 Solving Rational Equations Helpful Hint Extraneous solutions may be introduced by squaring both sides of an equation or by multiplying both sides of an equation by a variable expression. Holt Algebra 1 12-7 Solving Rational Equations Example 4: Extraneous Solutions Solve solutions. Step 1 Solve. . Identify any extraneous Use cross products. 2(x2 – 1) = (x + 1)(x – 6) Distribute 2 on the left side. Multiply the right side. 2x2 – 2 = x2 – 5x – 6 Subtract x2 from both sides. Add 5x and 6 to both sides. x2 + 5x + 4 = 0 Factor the quadratic expression. (x + 1)(x + 4) = 0 Use the Zero Product Property. Solve. x = –1 or x = –4 Holt Algebra 1 12-7 Solving Rational Equations Example 4 Continued Solve solutions. . Identify any extraneous Step 2 Find extraneous solutions. Because and are undefined –1 is not a solution. The only solution is – 4, so – 1 is an extraneous solution. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 4a Solve. Identify any extraneous solutions. Step 1 Solve. Use cross products. (x – 2)(x – 7) = 3(x – 7) Distribute 3 on the right side. Multiply the left side. 2x2 – 9x + 14 = 3x – 21 Subtract 3x from both sides. Add 21 to both sides. X2 – 12x + 35 = 0 Factor the quadratic expression. (x – 7)(x – 5) = 0 Use the Zero Product Property. Solve. x = 7 or x = 5 Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 4a Continued Step 2 Find extraneous solutions. Because and are undefined 7 is not a solution. The only solution is 5, so 7 is an extraneous solution. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 4b Solve. Identify any extraneous solutions. Step 1 Solve. (x + 1)(x – 3) = 4(x – 2) x2 – 2x – 3 = 4x – 8 X2 – 6x + 5 = 0 (x – 1)(x – 5) = 0 x = 1 or x = 5 Holt Algebra 1 Use cross products. Distribute 4 on the right side. Multiply the left side. Subtract 4x from both sides. Add 8 to both sides. Factor the quadratic expression. Use the Zero Product Property. Solve. 12-7 Solving Rational Equations Check It Out! Example 4b Continued Step 2 Find extraneous solutions. 1 and 5 are solutions. The solutions are 1 and 5, there are no extraneous solutions. Holt Algebra 1 12-7 Solving Rational Equations Check It Out! Example 4c Solve. Identify any extraneous solutions. Step 1 Solve. 6(x2 + 2x) = 9(x2) 3x2 – 12x = 0 3x(x – 4) = 0 3x = 0, or x – 4 = 0 x = 0 or x = 4 Holt Algebra 1 Use cross products. Distribute 6 on the left side. Multiply the right side. Subtract 9x2 from both sides. Multiply through with – 1. Factor the quadratic expression. Use the Zero Product Property. Solve. 12-7 Solving Rational Equations Check It Out! Example 4c Continued Step 2 Find extraneous solutions. Because and are undefined 0 is not a solution. The only solution is 4, so 0 is an extraneous solutions. Holt Algebra 1 12-7 Solving Rational Equations Lesson Quiz: Part I Solve each equation. Check your answer. 1. 24 2. –4, 3 3. 4. Pipe A can fill a tank with water in 4 hours. Pipe B can fill the same tank in 5 hours. How long will it take both pipes working together to fill the tank? Holt Algebra 1 12-7 Solving Rational Equations Lesson Quiz: Part II 5. Solve . Identify any extraneous solutions. –5; 3 is extraneous. Holt Algebra 1