Transcript 12-7
12-7Solving 12-7 SolvingRational RationalEquations Equations Objectives Notes Practice Holt Algebra Holt Algebra 11 12-7 Solving Rational Equations Objectives Solve rational equations. Identify extraneous solutions. 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 2 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 3: Solving Rational Equations by Using the LCD Solve the 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 3 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 3 Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Example 4: Solving Rational Equations by Using the LCD Solve the 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 4 Continued Step 3 Simplify and solve. Divide out common factors. 4x – 3 = x2 0 = x2 – 4x + 3 (x – 3)(x – 1) = 0 x = 3, 1 Holt Algebra 1 Simplify. Subtract 4x and -3 from both sides. Factor. Solve. 12-7 Solving Rational Equations Example 4 Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Example 5 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 Example 5 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 Example 5 Continued Check Verify that your solution is not extraneous. Holt Algebra 1 12-7 Solving Rational Equations Example 6: 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 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. 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 7: 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 7 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 Example 8 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 Example 8 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 • EXIT TICKET: – Solve the equation. 2 1 2 x x2 Holt Algebra 1