Holt Algebra Solving Rational Equations Warm Up 1. Find the LCM of x, 2x 2, and Find the LCM of p 2 – 4p and p 2 – 16. Multiply. Simplify your answer

Holt Algebra Solving Rational Equations Solve rational equations. Identify extraneous solutions. Objectives 11-8

Holt Algebra Solving Rational Equations rational equation Vocabulary 11-8

Holt Algebra 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. 11-8

Holt Algebra Solving Rational Equations 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 Check Distribute 3 on the right side. Subtract 3x from both sides. – 1 –

Holt Algebra 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 Check Divide both sides by

Holt Algebra 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. Check Divide both sides by

Holt Algebra 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. Check Divide both sides by

Holt Algebra 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. 11-8

Holt Algebra 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. 11-8

Holt Algebra Solving Rational Equations Example 2A Continued Step 3 Simplify and solve. (2x)(2) +6(x +1) = 5(x +1) 4x + 6x + 6 = 5x x + 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

Holt Algebra Solving Rational Equations Example 2A Continued Check Verify that your solution is not extraneous. 11-8

Holt Algebra Solving Rational Equations Example 2B: Solving Rational Equations by Using the LCD Solve each equation. Check your answer. Step 1 Find the LCD (x 2 ) Include every factor of the denominator. Step 2 Multiply both sides by the LCD Distribute on the left side. 11-8

Holt Algebra Solving Rational Equations 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, 1 Simplify. Subtract x 2 from both sides. Factor. Solve. Multiply by –

Holt Algebra Solving Rational Equations Example 2B Continued Check Verify that your solution is not extraneous. 11-8

Holt Algebra 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. 11-8

Holt Algebra 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. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 2a Continued Check Verify that your solution is not extraneous. 11-8

Holt Algebra 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. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 2b Continued Solve each equation. Check your answer. 12j – 10(2j + 4) = 4j j – 20j – 40 = 4j + 8 – 12j = 48 j = – 4 Simplify. Distribute 10. Combine like terms. Divide out common terms. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 2b Continued Check Verify that your solution is not extraneous. 11-8

Holt Algebra 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. 11-8

Holt Algebra 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 t 2 + 3t 0 = t 2 – 4t = (t – 3)(t – 1) t = 3, 1 Simplify. Distribute t. Combine like terms. Factor. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 2c Continued Check Verify that your solution is not extraneous. 11-8

Holt Algebra Solving Rational Equations When you multiply each side of an equation by the LCD, you may get an extraneous solution. Recall from previous chapters that an extraneous solution is a solution to a resulting equation that is not a solution to the original equation. 11-8

Holt Algebra Solving Rational Equations Extraneous solutions may be introduced by squaring both sides of an equation or by multiplying both sides of an equation by a variable expression. Helpful Hint 11-8

Holt Algebra Solving Rational Equations Example 4: Extraneous Solutions Solve. Identify any extraneous solutions. Step 1 Solve. 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. 11-8

Holt Algebra Solving Rational Equations Example 4 Continued Solve. Identify any extraneous solutions. 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. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 4a Solve. Identify any extraneous solutions. Step 1 Solve. (x – 2)(x – 7) = 3(x – 7) Use cross products. Distribute 3 on the right side. Multiply the left side. 2x 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. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 4a Continued Step 2 Find extraneous solutions. The only solution is 5, so 7 is an extraneous solution. Because and are undefined 7 is not a solution.  11-8

Holt Algebra Solving Rational Equations Check It Out! Example 4b Solve. Identify any extraneous solutions. Step 1 Solve. (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. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 4b Continued Step 2 Find extraneous solutions. The solutions are 1 and 5, there are no extraneous solutions. 1 and 5 are solutions. 11-8

Holt Algebra Solving Rational Equations Check It Out! Example 4c Solve. Identify any extraneous solutions. Step 1 Solve. 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 –

Holt Algebra Solving Rational Equations Check It Out! Example 4c Continued Step 2 Find extraneous solutions. The only solution is 4, so 0 is an extraneous solutions. Because and are undefined 0 is not a solution.  11-8

Holt Algebra Solving Rational Equations Lesson Quiz: Part I Solve each equation. Check your answer – 4,

Holt Algebra Solving Rational Equations Lesson Quiz: Part II 4. Solve. Identify any extraneous solutions. – 5; 3 is extraneous. 11-8