Solving Rational Equations Digital Lesson Solving Rational Equations

are rational expressions. For example, A rational expression is a fraction with polynomials for the numerator and denominator. are rational expressions. For example, If x is replaced by a number making the denominator of a rational expression zero, the value of the rational expression is undefined. Example: Evaluate for x = –3, 0, and 1. x  3 undefined undefined 9 undefined 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Rational Expression

A rational equation is an equation between rational expressions. For example, and are rational equations. To solve a rational equation: 1. Find the LCM of the denominators. 2. Clear denominators by multiplying both sides of the equation by the LCM. 3. Solve the resulting polynomial equation. 4. Check the solutions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Rational Equation

Examples: 1. Solve: . LCM = x – 3. 1 = x + 1 x = 0 (0) LCM = x(x – 1). Find the LCM. 1 = x + 1 Multiply by LCM = (x – 3). x = 0 Solve for x. (0) Check. Substitute 0. Simplify. True. 2. Solve: . LCM = x(x – 1). Find the LCM. Multiply by LCM. x – 1 = 2x Simplify. x = –1 Solve. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Examples: Solve

In this case, the value is not a solution of the rational equation. After clearing denominators, a solution of the polynomial equation may make a denominator of the rational equation zero. In this case, the value is not a solution of the rational equation. It is critical to check all solutions. Example: Solve: . Since x2 – 1 = (x – 1)(x + 1), LCM = (x – 1)(x + 1). 3x + 1 = x – 1 2x = – 2  x = – 1 Check. Since – 1 makes both denominators zero, the rational equation has no solutions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Solve

Find the LCD. Solve: 2 + 1 = 4 x 3 x 3x 3x ∙ 2 + 3x ∙ 1 = 3x ∙ 4 x 3 x Multiply each term by the LCD. 6 + x = 12 Simplify x = 6

Find the LCD. Solve: 6 + x = 4 x 2 2x 2x ∙ 6 + 2x ∙ x = 2x ∙ 4 x 2 Multiply each term by the LCD. 12 + x2 = 8x Simplify x2 – 8x + 12 = 0 (x – 6)(x – 2) = 0 x = 6, 2

Solve: 5 = y y + 2 3 Cross Multiply 15 = y(y + 2) 15 = y2 + 2y Simplify Write in standard form. Factor Solve

Solve: 3 = 2 . x + 5 x + 1 Cross Multiply 3(x + 1) = 2(x + 5) Simplify Combine Like-Terms to solve.

Example: Solve: . x2 – 8x + 15 = (x – 3)(x – 5) x(x – 5) = – 6 Factor. The LCM is (x – 3)(x – 5). Original Equation. x(x – 5) = – 6 Polynomial Equation. x2 – 5x + 6 = 0 Simplify. (x – 2)(x – 3) = 0 Factor. Check. x = 2 is a solution. x = 2 or x = 3 Check. x = 3 is not a solution since both sides would be undefined. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Solve