Add and Subtract Rational Expressions *

Transcription

1 OpenStax-CNX module: m Add and Subtract Rational Expressions * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of this section, you will be able to: Abstract Add and subtract rational expressions with a common denominator Add and subtract rational expressions whose denominators are opposites Find the least common denominator of rational expressions Add and subtract rational expressions with unlike denominators Add and subtract rational functions Before you get started, take this readiness quiz. 7 1.Add: If you missed this problem, review. 3x 2.Subtract: If you missed this problem, review. 3.Subtract: 6 (2x + 1) 4 (x 5). If you missed this problem, review. 1 Add and Subtract Rational Expressions with a Common Denominator What is the rst step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you nd one before you add. It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator. If p, q, and r are polynomials where r 0, then p r + q r = p + q r * Version 1.5: Apr 9, :00 am and p r q r = p q r (1)

2 OpenStax-CNX module: m To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator. We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors. Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example? Example 1 Add: 11x+28 x+4 + x2 x+4. Solution Since the denominator is x + 4, we must exclude the value x = 4. The fractions have a common denominator, so add the numerators and place the sum over the common denominator. 11x+28 x+4 + x2 x+4, x 4 11x+28+x 2 x+4 Write the degrees in descending order. x 2 +11x+28 x+4 Factor the numerator. (x+4)(x+7) x+4 Simplify by removing common factors. )(x+4)(x+7) )x+4 Simplify. x + 7 The expression simplies to x + 7 but the original expression had a denominator of x + 4 so x 4. Exercise 2 (Solution on p. 24.) Simplify: 9x+14 x+7 + x2 x+7. Exercise 3 (Solution on p. 24.) Simplify: x 2 +8x x x+5.

3 OpenStax-CNX module: m To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the dierence over the common denominator. Be careful of the signs when you subtract a binomial or trinomial. Example 2 Subtract: 5x 2 7x+3 x 2 3x+18 4x2 +x 9 x 2 3x+18. Solution 5x 2 7x+3 x 2 3x+18 4x2 +x 9 x 2 3x+18 Subtract the numerators and place the dierence over the common denominator. 5x 2 7x+3 (4x 2 +x 9) x 2 3x+18 Distribute the sign in the numerator. 5x 2 7x+3 4x 2 x+9 x 2 3x 18 Combine like terms. x 2 8x+12 x 2 3x 18 Factor the numerator and the denominator. (x 2)(x 6) (x+3)(x 6) Simplify by removing common factors. (x 2))(x 6) (x+3))(x 6) (x 2) (x+3) Exercise 5 (Solution on p. 24.) Subtract: 4x 2 11x+8 x 2 3x+2 3x2 +x 3 x 2 3x+2. Exercise 6 (Solution on p. 24.) Subtract: 6x 2 x+20 x x2 +11x 7 x 2 81.

4 OpenStax-CNX module: m Add and Subtract Rational Expressions Whose Denominators are Opposites When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by 1 1. Let's see how this works. Multiply the second fraction by 1 1. The denominators are the same. Simplify. Table 1 Be careful with the signs as you work with the opposites when the fractions are being subtracted. Example 3 Subtract: m 2 6m m 2 1 3m+2 1 m 2. Solution The denominators are opposites, so multiply the second fraction by 1 1. Simplify the second fraction. The denominators are the same. Subtract the numerators. continued on next page

5 OpenStax-CNX module: m Distribute. Combine like terms. Factor the numerator and denominator. Simplify by removing common factors. Simplify. Table 2 Exercise 8 (Solution on p. 24.) Subtract: y 2 5y y 2 4 6y 6 4 y 2. Exercise 9 (Solution on p. 24.) Subtract: 2n 2 +8n 1 n 2 1 n2 7n 1 1 n 2. 3 Find the Least Common Denominator of Rational Expressions When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions. 7 Let's look at this example: Since the denominators are not the same, the rst step was to nd the least common denominator (LCD). To nd the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we brought down one prime from each column. Finally, we multiplied the factors to nd the LCD. When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.

6 OpenStax-CNX module: m We do the same thing for rational expressions. However, we leave the LCD in factored form. Step 1.Factor each denominator completely. Step 2.List the factors of each denominator. Match factors vertically when possible. Step 3.Bring down the columns by including all factors, but do not include common factors twice. Step 4.Write the LCD as the product of the factors. Remember, we always exclude values that would make the denominator zero. What values of x should we exclude in this next example? Example 4 8 a Find the LCD for the expressions x 2 2x 3, expressions with the lowest common denominator. Solution a 3x x 2 +4x+3 and b rewrite them as equivalent rational 8 Find the LCD for x 2 2x 3, 3x x 2 +4x+3. Factor each denominator completely, common factors. lining up Bring down the columns. continued on next page

7 OpenStax-CNX module: m Write the LCD as the product of the factors. b Table 3 Factor each denominator. Multiply each denominator by the `missing' LCD factor and multiply each numerator by the same factor. Simplify the numerators. Table 4 Exercise 11 (Solution on p. 24.) 2 a Find the LCD for the expressions x 2 x 12, 1 expressions with the lowest common denominator. x 2 16 b rewrite them as equivalent rational Exercise 12 (Solution on p. 24.) 3x x 2 3x+10, 5 x 2 +3x+2 a Find the LCD for the expressions expressions with the lowest common denominator. b rewrite them as equivalent rational 4 Add and Subtract Rational Expressions with Unlike Denominators Now we have all the steps we need to add or subtract rational expressions with unlike denominators. Example 5: How to Add Rational Expressions with Unlike Denominators Add: 3 x x 2.

8 OpenStax-CNX module: m Solution Exercise 14 2 Add: x 2 + (Solution on p. 24.) 5 x+3. Exercise 15 4 Add: m+3 + (Solution on p. 24.) 3 m+4. The steps used to add rational expressions are summarized here. Step 1.Determine if the expressions have a common denominator. Yes go to step 2. No Rewrite each rational expression with the LCD. Find the LCD. Rewrite each rational expression as an equivalent rational Step 2.Add or subtract the rational expressions. Step 3.Simplify, if possible. expression with the LCD.

9 OpenStax-CNX module: m Avoid the temptation to simplify too soon. In the example above, we must leave the rst rational expression 2x 6 as to be able to add it to. Simplify only after you have combined the numerators. 3x 6 (x 3)(x 2) Example 6 8 Add: x 2 2x 3 + Solution 3x x 2 +4x+3. (x 2)(x 3) Do the expressions have a common denominator? Rewrite each expression with the LCD. No. Find the LCD. x 2 2x 3 = (x + 1) (x 3) x 2 + 4x + 3 = (x + 1) (x + 3) LCD = (x + 1) (x 3) (x + 3) continued on next page

10 OpenStax-CNX module: m Rewrite each rational expression as an equivalent rational expression with the LCD. Simplify the numerators. Add the rational expressions. Simplify the numerator. Table 5 The numerator is prime, so there are no common factors. Exercise 17 (Solution on p. 24.) Add: 1 m 2 m 2 + 5m m 2 +3m+2. Exercise 18 (Solution on p. 24.) Add: 2n n 2 3n n 2 +5n+6. The process we use to subtract rational expressions with dierent denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators. Example 7 8y Subtract: y y 4. Solution

11 OpenStax-CNX module: m Do the expressions have a common denominator? Rewrite each expression with the LCD. Find the LCD. y 2 16 = (y 4) (y + 4) y 4 = y 4 LCD = (y 4) (y + 4) Rewrite each rational expression as an equivalent rational expression with the LCD. No. Simplify the numerators. Subtract the rational expressions. Simplify the numerator. Factor the numerator to look for common factors. continued on next page

12 OpenStax-CNX module: m Remove common factors Simplify. Table 6 Exercise 20 (Solution on p. 24.) Subtract: 2x x x+2. Exercise 21 (Solution on p. 24.) Subtract: 3 z+3 6z z 2 9. There are lots of negative signs in the next example. Be extra careful. Example 8 Subtract: 3n 9 n 2 +n 6 n+3 2 n. Solution Factor the denominator. Since n 2 and 2 n are opposites, we will multiply the second rational expression by 1 1. continued on next page

13 OpenStax-CNX module: m Simplify. Remember, a ( b) = a + b. Do the rational expressions have a common denominator? No. Find the LCD. n 2 + n 6 = (n 2) (n + 3) n 2 = (n 2) LCD = (n 2) (n + 3) Rewrite each rational expression as an equivalent rational expression with the LCD. Simplify the numerators. Add the rational expressions. Simplify the numerator. Factor the numerator to look for common factors. Simplify. Table 7 Exercise 23 (Solution on p. 24.) Subtract : 3x 1 x 2 5x x.

14 OpenStax-CNX module: m Exercise 24 (Solution on p. 24.) Subtract: 2y 2 y 2 +2y 8 y 1 2 y. Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator. Example 9 Subtract: 4 a 2 +6a+5 3 a 2 +7a+10. Solution Factor the denominators. Do the rational expressions have a common denominator? No. Find the LCD. a 2 + 6a + 5 = (a + 1) (a + 5) a 2 + 7a + 10 = (a + 5) (a + 2) Rewrite each rational expression as an equivalent rational expression with the LCD. LCD = (a + 1) (a + 5) (a + 2) Simplify the numerators. Subtract the rational expressions. continued on next page

15 OpenStax-CNX module: m Simplify the numerator. Look for common factors. Simplify. Table 8 Exercise 26 (Solution on p. 24.) Subtract: 3 b 2 4b 5 2 b 2 6b+5. Exercise 27 (Solution on p. 24.) Subtract: 4 x x 2 x 2. We follow the same steps as before to nd the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to nd their LCD. Example 10 2u Simplify: u u 2u 1 u 2 u. Solution continued on next page

16 OpenStax-CNX module: m Do the expressions have a common denominator? No. Rewrite each expression with the LCD. Find the LCD. u 1 = (u 1) u = u u 2 u = u (u 1) LCD = u (u 1) Rewrite each rational expression as an equivalent rational expression with the LCD. Write as one rational expression. Simplify. Factor the numerator, and remove common factors. Simplify. Table 9 Exercise 29 (Solution on p. 24.) Simplify: v v v 1 6 v 2 1. Exercise 30 (Solution on p. 24.) Simplify: 3w w w+7 17w+4 w 2 +9w+14.

17 OpenStax-CNX module: m Add and subtract rational functions To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions. Example 11 Find R (x) = f (x) g (x) where f (x) = x+5 x 2 Solution and g (x) = 5x+18 x 2 4. Substitute in the functions f (x),g (x). Factor the denominators. Do the expressions have a common denominator? No. Rewrite each expression with the LCD. Find the LCD. x 2 = (x 2) x 2 4 = (x 2) (x + 2) LCD = (x 2) (x + 2) Rewrite each rational expression as an equivalent rational expression with the LCD. Write as one rational expression. continued on next page

18 OpenStax-CNX module: m Simplify. Factor the numerator, and remove common factors. Simplify. Table 10 Exercise 32 (Solution on p. 24.) Find R (x) = f (x) g (x) where f (x) = x+1 x+3 and g (x) = x+17 x 2 x 12. Exercise 33 (Solution on p. 24.) Find R (x) = f (x) + g (x) where f (x) = x 4 x+3 and g (x) = 4x+6 x 2 9. Access this online resource for additional instruction and practice with adding and subtracting rational expressions. Add and Subtract Rational Expressions- Unlike Denominators 1 6 Key Concepts ˆ Rational Expression Addition and Subtraction If p, q, and r are polynomials where r 0, then p r + q r = p+q r and p r q r = p q r ˆ How to nd the least common denominator of rational expressions. Step a. Factor each expression completely. Step b. List the factors of each expression. Match factors vertically when possible. Step c. Bring down the columns. Step d. Write the LCD as the product of the factors. 1

19 OpenStax-CNX module: m ˆ How to add or subtract rational expressions. Step a. Determine if the expressions have a common denominator. ˆ Yes go to step 2. ˆ No Rewrite each rational expression with the LCD. ˆ Find the LCD. ˆ Rewrite each rational expression as an equivalent rational expression with the LCD. Step b. Add or subtract the rational expressions. Step c. Simplify, if possible Practice Makes Perfect Add and Subtract Rational Expressions with a Common Denominator In the following exercises, add. Exercise 34 (Solution on p. 25.) Exercise Exercise 36 (Solution on p. 25.) 3c 4c c 5 Exercise 37 7m 2m+n + 4 2m+n Exercise 38 (Solution on p. 25.) 2r 2 2r r 8 2r 1 Exercise 39 3s 2 3s s 10 3s 2 Exercise 40 (Solution on p. 25.) 2w 2 w w w 2 16 Exercise 41 7x 2 x x x 2 9 In the following exercises, subtract. Exercise 42 (Solution on p. 25.) 9a 2 3a a 7 Exercise 43 25b 2 5b b 6 Exercise 44 (Solution on p. 25.) 3m 2 6m 30 21m 30 6m 30 Exercise 45 2n 2 4n 32 18n 16 4n 32 Exercise 46 (Solution on p. 25.) 6p 2 +3p+4 p 2 +4p 5 Exercise 47 5q 2 +3q 9 q 2 +6q+8 5p2 +p+7 p 2 +4p 5 4q2 +9q+7 q 2 +6q+8

20 OpenStax-CNX module: m Exercise 48 (Solution on p. 25.) 5r 2 +7r 33 r 2 49 Exercise 49 7t 2 t 4 t r2 +5r+30 r t2 +12t 44 t 2 25 Add and Subtract Rational Expressions whose Denominators are Opposites In the following exercises, add or subtract. Exercise 50 (Solution on p. 25.) 10v 2v 1 + 2v+4 1 2v Exercise 51 20w 5w 2 + 5w+6 2 5w Exercise 52 (Solution on p. 25.) 10x 2 +16x 7 8x 3 Exercise 53 6y 2 +2y 11 3y 7 + 2x2 +3x 1 3 8x + 3y2 3y y Exercise 54 (Solution on p. 25.) z 2 +6z z z z 2 Exercise 55 a 2 +3a a 2 9 3a 27 9 a 2 Exercise 56 (Solution on p. 25.) 2b 2 +30b 13 b 2 49 Exercise 57 c 2 +5c 10 c b2 5b 8 49 b 2 c2 8c c 2 Find the Least Common Denominator of Rational Expressions In the following exercises, a nd the LCD for the given rational expressions b rewrite them as equivalent rational expressions with the lowest common denominator. Exercise 58 (Solution on p. 25.) 5 x 2 2x 8, Exercise 59 2x x 2 x 12 8 y 2 +12y+35, 3y y 2 +y 42 Exercise 60 (Solution on p. 25.) 9 z 2 +2z 8, 4z z 2 4 Exercise 61 6 a 2 +14a+45, 5a a 2 81 Exercise 62 (Solution on p. 25.) 4 b 2 +6b+9, 2b b 2 2b 15 Exercise 63 5 c 2 4c+4, 3c c 2 7c+10 Exercise 64 (Solution on p. 25.) 2 3d 2 +14d 5, 5d 3d 2 19d+6 Exercise m 2 3m 2, 6m 5m 2 +17m+6 Add and Subtract Rational Expressions with Unlike Denominators In the following exercises, perform the indicated operations. Exercise 66 (Solution on p. 25.) 7 10x 2 y xy 2

21 OpenStax-CNX module: m Exercise a 3 b a 2 b 3 Exercise 68 (Solution on p. 25.) 3 r r 5 Exercise 69 4 s s+3 Exercise 70 (Solution on p. 25.) 5 3w w+1 Exercise x x 1 Exercise 72 (Solution on p. 25.) 2y y y 1 Exercise 73 3z z z+5 Exercise 74 (Solution on p. 26.) 5b a 2 b 2a 2 + 2b b 2 4 Exercise 75 4 cd+3c + 1 d 2 9 Exercise 76 (Solution on p. 26.) 3m 3m 3 + 5m m 2 +3m 4 Exercise n n 2 n 2 Exercise 78 (Solution on p. 26.) 3r r 2 +7r r 2 +4r+3 Exercise 79 2s s 2 +2s s 2 +3s 10 Exercise 80 (Solution on p. 26.) t t 6 t 2 t+6 Exercise 81 x 3 x+6 x x+3 Exercise 82 (Solution on p. 26.) 5a a+3 a+2 a+6 Exercise 83 3b b 2 b 6 b 8 Exercise 84 (Solution on p. 26.) 6 m+6 12m m 2 36 Exercise 85 4 n+4 8n n 2 16 Exercise 86 (Solution on p. 26.) 9p 17 p 2 4p 21 p+1 7 p Exercise 87 13q 8 q 2 +2q 24 q+2 4 q Exercise 88 (Solution on p. 26.) 2r 16 r 2 +6r r

22 OpenStax-CNX module: m Exercise 89 2t 30 t 2 +6t t Exercise 90 (Solution on p. 26.) 2x+7 10x Exercise 91 8y 4 5y+2 6 Exercise 92 (Solution on p. 26.) 3 x 2 3x 4 2 x 2 5x+4 Exercise 93 4 x 2 6x+5 3 x 2 7x+10 Exercise 94 (Solution on p. 26.) 5 x 2 +8x 9 4 x 2 +10x+9 Exercise x 2 +5x+2 1 2x 2 +3x+1 Exercise 96 (Solution on p. 26.) 5a a a 2a+18 a 2 2a Exercise 97 2b b b 2b 15 2b 2 10b Exercise 98 (Solution on p. 26.) c c c 2 10c c 2 4 Exercise 99 6d d d+4 7d 5 d 2 d 20 Exercise 100 (Solution on p. 26.) 3d d d d+8 d 2 +2d Exercise 101 2q q q 3 13q+15 q 2 +2q 15 Add and Subtract Rational Functions In the following exercises, nd ar (x) = f (x) + g (x)br (x) = f (x) g (x). Exercise 102 (Solution on p. 26.) f (x) = 5x 5 x 2 +x 6 and g (x) = x+1 2 x Exercise 103 f (x) = 4x 24 x 2 +x 30 and g (x) = x+7 5 x Exercise 104 (Solution on p. 26.) f (x) = 6x x 2 64 and g (x) = 3 x 8 Exercise 105 f (x) = 5 g (x) = x+7 and 10x x 2 49

23 OpenStax-CNX module: m Writing Exercises Exercise 106 (Solution on p. 26.) Donald thinks that 3 x + 4 x is 7 2x. Is Donald correct? Explain. Exercise 107 Explain how you nd the Least Common Denominator of x 2 + 5x + 4 and x Exercise 108 (Solution on p. 26.) Felipe thinks 1 x + 1 y is 2 x+y. a Choose numerical values for x and y and evaluate 1 x + 1 y. 2 b Evaluate x+y for the same values of x and y you used in part a. c Explain why Felipe is wrong. d Find the correct expression for 1 x + 1 y. Exercise Simplify the expression n 2 +6n+9 1 n 2 9 and explain all your steps. 7.3 Self Check a After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. b After reviewing this checklist, what will you do to become condent for all objectives?

24 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. 2) x + 2 Solution to Exercise (p. 2) x + 3 Solution to Exercise (p. 3) x 11 x 2 Solution to Exercise (p. 3) x 3 x+9 Solution to Exercise (p. 5) y+3 y+2 Solution to Exercise (p. 5) 3n 2 n 1 Solution to Exercise (p. 7) a(x 4) (x + 3) (x + 4) b 2x+8 (x 4)(x+3)(x+4), x+3 (x 4)(x+3)(x+4) Solution to Exercise (p. 7) a(x + 2) (x 5) (x + 1) b 3x 2 +3x (x+2)(x 5)(x+1), 5x 25 (x+2)(x 5)(x+1) Solution to Exercise (p. 8) 7x 4 (x 2)(x+3) Solution to Exercise (p. 8) 7m+25 (m+3)(m+4) Solution to Exercise (p. 10) 5m 2 9m+2 (m+1)(m 2)(m+2) Solution to Exercise (p. 10) 2n 2 +12n 30 (n+2)(n 5)(n+3) Solution to Exercise (p. 12) 1 x 2 Solution to Exercise (p. 12) 3 z 3 Solution to Exercise (p. 13) 5x+1 (x 6)(x+1) Solution to Exercise (p. 14) y+3 y+4 Solution to Exercise (p. 15) 1 (b+1)(b 1) Solution to Exercise (p. 15) 1 (x+2)(x+1) Solution to Exercise (p. 16) v+3 v+1 Solution to Exercise (p. 16) 3w w+7 Solution to Exercise (p. 18) x 7 x 4

25 OpenStax-CNX module: m Solution to Exercise (p. 18) x 2 3x+18 (x+3)(x 3) Solution to Exercise (p. 19) 3 5 Solution to Exercise (p. 19) 3c+5 4c 5 Solution to Exercise (p. 19) r + 8 Solution to Exercise (p. 19) 2w w 4 Solution to Exercise (p. 19) 3a + 7 Solution to Exercise (p. 19) m 2 2 Solution to Exercise (p. 19) p+3 p+5 Solution to Exercise (p. 19) r+9 r+7 Solution to Exercise (p. 20) 4 Solution to Exercise (p. 20) x + 2 Solution to Exercise (p. 20) z+4 z 5 Solution to Exercise (p. 20) 4b 3 b 7 Solution to Exercise (p. 20) a(x + 2) (x 4) (x + 3) 5x+15 b (x+2)(x 4)(x+3), 2x 2 +4x (x+2)(x 4)(x+3) Solution to Exercise (p. 20) a(z 2) (z + 4) (z 4) 9z 36 b (z 2)(z+4)(z 4), 4z 2 8z (z 2)(z+4)(z 4) Solution to Exercise (p. 20) a(b + 3) (b + 3) (b 5) 4b 20 b (b+3)(b+3)(b 5), 2b 2 +6b (b+3)(b+3)(b 5) Solution to Exercise (p. 20) a(d + 5) (3d 1) (d 6) b 2d 12 (d+5)(3d 1)(d 6), 5d 2 +25d (d+5)(3d 1)(d 6) Solution to Exercise (p. 20) 21y+8x 30x 2 y 2 Solution to Exercise (p. 21) 5r 7 (r+4)(r 5) Solution to Exercise (p. 21) 11w+1 (3w 2)(w+1)

26 OpenStax-CNX module: m Solution to Exercise (p. 21) 2y 2 +y+9 (y+3)(y 1) Solution to Exercise (p. 21) b(5b+10+2a 2 ) a 2 (b 2)(b+2) Solution to Exercise (p. 21) m m+4 Solution to Exercise (p. 21) 3(r 2 +6r+18) (r+1)(r+6)(r+3) Solution to Exercise (p. 21) 2(7t 6) (t 6)(t+6) Solution to Exercise (p. 21) 4a 2 +25a 6 (a+3)(a+6) Solution to Exercise (p. 21) 6 m 6 Solution to Exercise (p. 21) p+2 p+3 Solution to Exercise (p. 21) 3 r 2 Solution to Exercise (p. 22) 4(8x+1) 10x 1 Solution to Exercise (p. 22) x 5 (x 4)(x+1)(x 1) Solution to Exercise (p. 22) 1 (x 1)(x+1) Solution to Exercise (p. 22) 5a 2 +7a 36 a(a 2) Solution to Exercise (p. 22) c 5 c+2 Solution to Exercise (p. 22) 3(d+1) d+2 Solution to Exercise (p. 22) a R (x) = (x+8)(x+1) (x 2)(x+3) Solution to Exercise (p. 22) a 3(3x+8) (x 8)(x+8) b R (x) = x+1 x+3 b R (x) = 3 x+8 Solution to Exercise (p. 23) Answers will vary. Solution to Exercise (p. 23) a Answers will vary. b Answers will vary. c Answers will vary. d x+y xy