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10-7 Solving Rational Equations Preview Warm Up California Standards Lesson Presentation 10-7 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. 4. 10-7 Solving Rational Equations California Standards Preparation for 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. 10-7 Solving Rational Equations Vocabulary rational equation extraneous solutions 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. 10-7 Solving Rational Equations Additional Example 1: Solving Rational Equations by Using Cross Products Solve . Check your answer. 5x = (x – 2)(3) Use cross products. 5x = 3x – 6 Distribute 3 on the right side. 2x = –6 Subtract 3x from both sides. x = –3 Divide both sides by 2. 10-7 Solving Rational Equations Additional Example 1 Continued Check Substitute –3 for x in the original equation. –1 –1 10-7 Solving Rational Equations Check It Out! Example 1a Solve . Check your answer. 3n = (n + 4)(1) Use cross products. 3n = n + 4 Distribute 1 on the right side. 2n = 4 Subtract n from both sides. n=2 Divide both sides by 2. 10-7 Solving Rational Equations Check It Out! Example 1a Continued Check Substitute 2 for n in the original equation. 10-7 Solving Rational Equations Check It Out! Example 1b Solve . Check your answer. 4h = (h + 1)(2) Use cross products. 4h = 2h + 2 Distribute 2 on the right side. 2h = 2 Subtract 2h from both sides. h=1 Divide both sides by 2. 10-7 Solving Rational Equations Check It Out! Example 1b Continued Check Substitute 1 for h in the original equation. 10-7 Solving Rational Equations Check It Out! Example 1c Solve . Check your answer. 21x = (x – 7)(3) Use cross products. 21x = 3x – 21 Distribute 3 on the right side. 18x = –21 Subtract 3x from both sides. x= Divide both sides by 18. 10-7 Solving Rational Equations Check It Out! Example 1c Continued Check Substitute for x in the original equation. 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. 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. 10-7 Solving Rational Equations Additional Example 2A Continued Step 3 Simplify and solve. Divide out common factors. (2x)(2) + 6(x + 1) = 5(x + 1) Simplify. 4x + 6x + 6 = 5x + 5 Distribute and multiply. 10x + 6 = 5x + 5 Combine like terms. Subtract 5x and 6 from both sides. Divide both sides by 5. 5x = –1 10-7 Solving Rational Equations Additional Example 2A Continued Check 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) Include every factor of the denominator. Step 2 Multiply both sides by the LCD. Distribute on the left side. 10-7 Solving Rational Equations Additional 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. 10-7 Solving Rational Equations Additional Example 2B Continued Check 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. 10-7 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 Distribute the 4. –4 = a Subtract the 4 and 3a from both sides. 10-7 Solving Rational Equations Check It Out! Example 2a Continued Check 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. 10-7 Solving Rational Equations Check It Out! Example 2b Continued Solve each equation. Check your answer. Divide out common factors. 12j – 10(2j + 4) = 4j + 8 12j – 20j – 40 = 4j + 8 –12j = 48 j = –4 Simplify. Distribute 10. Combine like terms. Divide both sides by –12. 10-7 Solving Rational Equations Check It Out! Example 2b Continued Check 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. 10-7 Solving Rational Equations Additional Example 3: Extraneous Solutions Solve . Check your answer. 2(x2 – 1) = (x + 1)(x – 6) Use cross products. Distribute 2 on the left side. 2x2 – 2 = x2 – 5x – 6 Multiply the right side. Subtract x2 from both sides. x2 + 5x + 4 = 0 Add 5x and 6 to both sides. Factor the quadratic expression. (x + 1)(x + 4) = 0 x = –1 or x = –4 Use the Zero Product Property. Solve. 10-7 Solving Rational Equations Additional Example 3 Continued Solve . Check your answer. Check Because and are undefined, –1 is not a solution. –1 is an extraneous solution. The only solution is –4. 10-7 Solving Rational Equations Check It Out! Example 3a Solve. Check your answer. (x – 2)(x – 7) = 3(x – 7) Use cross products. x2 – 9x + 14 = 3x – 21 Distribute 3 on the right side. Multiply the left side. x2 – 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. 10-7 Solving Rational Equations Check It Out! Example 3a Continued Check Because and are undefined, 7 is not a solution. 7 is an extraneous solution. The only solution is 5. 10-7 Solving Rational Equations Check It Out! Example 3b Solve. Check your answer. (x + 1)(x – 3) = 4(x – 2) Use cross products. x2 – 2x – 3 = 4x – 8 Distribute 4 on the right side. Multiply the left side. x2 – 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. 10-7 Solving Rational Equations Check It Out! Example 3b Continued Check 1 and 5 are solutions. There are no extraneous solutions. The solutions are 1 and 5. 10-7 Solving Rational Equations Check It Out! Example 3c Solve. Identify any extraneous solutions. 6(x2 + 2x) = 9(x2) 6x2 + 12x) = 9x2 3x2 – 12x = 0 3x(x – 4) = 0 3x = 0, or x – 4 = 0 x = 0 or x = 4 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. 10-7 Solving Rational Equations Check It Out! Example 3c Continued Check Because and are undefined, 0 is not a solution. 0 is an extraneous solution. The only solution is 4. 10-7 Solving Rational Equations Lesson Quiz Solve each equation. Check your answer. 1. 24 –4, 3 2. 3. 4. Solve Check your answer. –5 .