1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8.4 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8.4 - 2 Rational Expressions and Functions Chapter 8

3. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 8.4 - 3 8.4 Equations with Rational Expressions and Graphs

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 4 8.4 Equations with Rational Expressions and Graphs Objectives 1. Determine the domain of the variable in a rational expression. 2. Solve rational equations. 3. Recognize the graph of a rational function.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 5 Find the domain of the equation. (a) 5 x = 1 6 – 11 2x2x EXAMPLE 1 Determining the Domains of Rational Equations 8.4 Equations with Rational Expressions and Graphs The domain is { x | x ≠ 0 }. The domains of the three rational terms of the equation are, in order, { x | x ≠ 0 }, (- ∞, ∞ ), { x | x ≠ 0 }. The intersection of these three domains is all real numbers except 0, which may be written { x | x ≠ 0 }.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 6 (b) 3 x – 1 = 2 x + 1 – 6 x 2 – 1 Find the domain of the equation. EXAMPLE 1 Determining the Domains of Rational Equations 8.4 Equations with Rational Expressions and Graphs The domains of the three rational terms are, respectively, { x | x ≠ 1 }, { x | x ≠ –1 }, { x | x ≠ + 1 }. The domain of the equation is the intersection of the three domains, all real numbers except 1 and –1, written { x | x ≠ + 1 }. The domain is { x | x ≠ + 1 }.

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 7 8.4 Equations with Rational Expressions and Graphs Caution on “Solutions” CAUTION When each side of an equation is multiplied by a variable expression, the resulting “solutions” may not satisfy the original equation. You must either determine and observe the domain or check all potential solutions in the original equation. It is wise to do both.

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 8 Solve. 5 x = 1 6 – 11 2x2x EXAMPLE 2 Solving an Equation with Rational Expressions 8.4 Equations with Rational Expressions and Graphs The domain, which excludes 0, was found in Example 1(a). = 5 x 1 6 – 11 2x2x 6x6x 6x6x Multiply by the LCD, 6 x. = 5 x 1 6 – 11 2x2x 6x6x 6x6x 6x6x Distributive property =30 x –33Multiply. = x –3Subtract 30. = x – 3Divide by –1.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 9 Solve. 5 x = 1 6 – 11 2x2x EXAMPLE 2 Solving an Equation with Rational Expressions 8.4 Equations with Rational Expressions and Graphs Check: Replace x with –3 in the original equation. = 5 –3 1 6 – 11 2(–3) Let x = –3. = 5 x 1 6 – 11 2x2x Original equation ? = 10 6 1 6 – 11 –6 – ? = 11 6 6 –– True The solution is { –3 }.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 10 Using the result from Example 1(b), we know that the domain excludes 1 and –1, since these values make one or more of the denominators in the equation equal 0. EXAMPLE 3 Solving an Equation with No Solution 8.4 Equations with Rational Expressions and Graphs Solve. 3 x – 1 = 2 x + 1 – 6 x 2 – 1

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 11 EXAMPLE 3 Solving an Equation with No Solution 8.4 Equations with Rational Expressions and Graphs Distributive property Solve. 3 x – 1 = 2 x + 1 – 6 x 2 – 1 = 3 x – 1 2 x + 1 – 6 x 2 – 1 ( x – 1)( x + 1) = 3 x – 1 2 x + 1 – 6 x 2 – 1 ( x – 1)( x + 1) = – 63( x + 1)2( x – 1) = – 63 x + 32 x + 2 = 6 x + 5 = 1 x Multiply each side by the LCD, ( x –1)( x + 1). Multiply. Distributive property Combine terms. Subtract 5.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 12 EXAMPLE 3 Solving an Equation with No Solution 8.4 Equations with Rational Expressions and Graphs Solve. 3 x – 1 = 2 x + 1 – 6 x 2 – 1 Since 1 is not in the domain, it cannot be a solution of the equation. Substituting 1 in the original equation shows why. Check: = 3 x – 1 2 x + 1 – 6 x 2 – 1 = 3 1 – 1 2 1 + 1 – 6 1 2 – 1 = 3 0 2 2 – 6 0 Since division by 0 is undefined, the given equation has no solution, and the solution set is ∅.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 13 Solve. 4 a 2 – 9 = 3 2( a 2 – 2 a – 3) – 6 a 2 + 4 a + 3 EXAMPLE 4 Solving an Equation with Rational Expressions 8.4 Equations with Rational Expressions and Graphs Factor each denominator to find the LCD, 2( a + 3)( a – 3)( a + 1). The domain excludes –3, 3, and –1.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 14 Multiply each side by the LCD, 2( a + 3)( a – 3)( a + 1). EXAMPLE 4 Solving an Equation with Rational Expressions 8.4 Equations with Rational Expressions and Graphs Solve. 4 a 2 – 9 = 3 2( a 2 – 2 a – 3) – 6 a 2 + 4 a + 3 = 4 ( a + 3)( a – 3) 3 2( a – 3)( a + 1) – 6 ( a + 3)( a + 1) 2( a + 3)( a – 3)( a + 1) 4 · 2( a + 1) 3( a + 3) – 6 · 2 ( a – 3) = 8 a + 83 a – 9 – 12 a – 36= 5 a – 1 12 a – 36= 35 7a7a = Distributive property Combine terms Subtract 5 a ; Add 36. Distributive property 5 a = Divide by 7.

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 15 Note that 5 is in the domain; substitute 5 in for a in the original equation to check that the solution set is { 5 }. EXAMPLE 4 Solving an Equation with Rational Expressions 8.4 Equations with Rational Expressions and Graphs Solve. 4 a 2 – 9 = 3 2( a 2 – 2 a – 3) – 6 a 2 + 4 a + 3 5 a =

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 16 Since the denominator cannot equal 0, – is excluded from the domain, as is 0. 1 2 EXAMPLE 5 Solving an Equation That Leads to a Quadratic Equation 8.4 Equations with Rational Expressions and Graphs Solve. 4 2 x + 1 – 2 x = 8x8x

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 17 EXAMPLE 5 Solving an Equation That Leads to a Quadratic Equation 8.4 Equations with Rational Expressions and Graphs Solve. 4 2 x + 1 – 2 x = 8x8x Multiply each side by the LCD, x (2 x + 1). 4 2 x + 1 = – 2 x 8x8x x (2 x + 1) 4x4x =–2(2 x + 1) 8x28x2 4x4x =–4 x + 2 8x28x2 0=–8 x 2 + 2 0=–2(4 x 2 – 1) 0=–2(2 x + 1)(2 x – 1) Distributive property Subtract 4 x ; standard form. Factor.

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 18 Because – is not in the domain of the equation, it is not a solution. Check that the solution set is. 1 2 1 2 EXAMPLE 5 Solving an Equation That Leads to a Quadratic Equation 8.4 Equations with Rational Expressions and Graphs Solve. 4 2 x + 1 – 2 x = 8x8x 0=–2(2 x + 1)(2 x – 1) 2 x + 1 = 0 or 2 x – 1 = 0 Zero-factor property x = – or x = 1 2 1 2

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 19 8.4 Equations with Rational Expressions and Graphs The domain of this function includes all real numbers except x = 0. Thus, there will be no point on the graph with x = 0. The vertical line with equation x = 0 is called a vertical asymptote of the graph. Graph of f ( x ) = 1 x

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 20 8.4 Equations with Rational Expressions and Graphs Graph of f ( x ) = 1 x The horizontal line with equation y = 0 is called a horizontal asymptote. Notice the closer positive values of x are to 0, the larger y is. Similarly, the closer negative values of x are to 0, the smaller (more negative) y is. Plot several points to verify this graph.

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 21 There is no point on the graph for x = 3 because 3 is excluded from the domain. The dashed line x = 3 represents the vertical asymptote and is not part of the graph. Notice the graph gets closer to the vertical asymptote as the x -values get closer to 3. 8.4 Equations with Rational Expressions and Graphs Graph of g ( x ) = –2 x – 3

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 22 Observe the y -values as the x -values get closer to the vertical asymptote (from both sides). As the x -values get closer to the vertical asymptote from the left, the y -values get larger and as the x -values get closer to the vertical asymptote from the right, the y -values get smaller (more negative). 8.4 Equations with Rational Expressions and Graphs Graph of g ( x ) = –2 x – 3

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.4 - 23 Again, y = 0 is a horizontal asymptote. Plot several points to verify this graph. 8.4 Equations with Rational Expressions and Graphs Graph of g ( x ) = –2 x – 3