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Simplifying Rational Expressions

15 24. –. Simplifying Rational Expressions. ALGEBRA 1 LESSON 12-3. (For help, go to Lessons 9-5 and 9-6.). Write each fraction in simplest form. 8 2. 25 35. 1. 2. 3. Factor each quadratic expression. 4. x 2 + x – 12 5. x 2 + 6 x + 8 6. x 2 – 2 x – 15.

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Simplifying Rational Expressions

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  1. 15 24 – Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 (For help, go to Lessons 9-5 and 9-6.) Write each fraction in simplest form. 8 2 25 35 1. 2. 3. Factor each quadratic expression. 4.x2 + x – 12 5.x2 + 6x + 8 6.x2 – 2x – 15 7.x2 + 8x + 16 8.x2 – x – 12 9.x2 – 7x + 12 12-3

  2. 5 • 5 5 • 7 3 • 5 3 • 8 25 35 5 7 3. = = 15 24 – Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 Solutions 8 2 5 8 1. = 8 ÷ 2 = 4 2. = – = – 4. Factors of –12 with a sum of 1: 4 and –3.x2 + x – 12 = (x + 4)(x – 3) 5. Factors of 8 with a sum of 6: 2 and 4.x2 + 6x + 8 = (x + 2)(x + 4) 6. Factors of –15 with a sum of –2: 3 and –5.x2 – 2x – 15 = (x + 3)(x – 5) 12-3

  3. Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 Solutions (continued) 7. Factors of 16 with a sum of 8: 4 and 4.x2 + 8x + 16 = (x + 4)(x + 4) or (x + 4)2 8. Factors of –12 with a sum of –1: 3 and –4.x2 – x – 12 = (x + 3)(x – 4) 9. Factors of 12 with a sum of –7: –3 and –4.x2 – 7x + 12 = (x – 3)(x – 4) 12-3

  4. Factor the numerator. The denominator cannot be factored. 3x + 9 x + 3 3(x + 3) x + 3 = = Divide out the common factor x + 3. 1 3(x + 3) x + 3 1 Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 3x + 9 x + 3 Simplify . = 3 Simplify. 12-3

  5. 4x – 20 x2 – 9x + 20 = Factor the numerator and the denominator. 4(x – 5) (x – 4) (x – 5) 4(x – 5) (x – 4) (x – 5) 4x – 20 x2 – 9x + 20 = Divide out the common factor x – 5. 1 1 = Simplify. 4 x – 4 Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 Simplify . 12-3

  6. 3x – 27 81 – x2 3(x – 9) (9 – x) (9 + x) = Factor the numerator and the denominator. 3(x – 9) – 1 (x – 9) (9 + x) = Factor –1 from 9 – x. = Divide out the common factor x – 9. 1 3(x – 9) – 1 (x – 9) (9 + x) 1 3 9 + x = – Simplify. Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 3x – 27 81 – x2 Simplify . 12-3

  7. = Substitute 8 for r and 3 for h. 30rh r + h 720 11 = Simplify. 60 • volume surface area 30rh r + h 30 (3) (8) 3 + 8 Round to the nearest whole number. 65 Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 The baking time for bread depends, in part, on its size and shape. A good approximation for the baking time, in minutes, of a cylindrical loaf is , or , where the radius r and the length h of the baked loaf are in inches. Find the baking time for a loaf that is 8 inches long and has a radius of 3 inches. Round your answer to the nearest minute. The baking time is approximately 65 minutes. 12-3

  8. x2 + 8x x2 – 64 –1 x + 6 x x – 8 6x2 – x – 12 8x2 – 10x – 3 x – 6 36 – x2 4x2 + x x3 Simplifying Rational Expressions ALGEBRA 1 LESSON 12-3 Simplify each expression. 3. 1. 2. 6 – 2x x – 3 –2 4. 5. 4x + 1 x2 3x + 4 4x + 1 12-3

  9. Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 (For help, go to Lessons 8-3 and 9-6.) Simplify each expression. 1.r2 • r82.b3 • b43.c7 ÷ c2 4. 3x4 • 2x55. 5n2 • n26. 15a3 (–3a2) Factor each polynomial. 7. 2c2 + 15c + 7 8. 15t2 – 26t + 11 9. 2q2 + 11q + 5 12-4

  10. Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 Solutions 1.r2 • r8 – r(2 + 8) = r102.b3 • b4 = b(3+4) = b7 3.c7÷c2 = c(7 – 2) = c54. 3x4 • 2x5 = (3 • 2)(x4 • x5) = 6x(4 + 5) = 6x9 5. 5n2 • n2 = 5(n2 • n2) = 5n(2 + 2) = 5n4 6. 15a3(–3a2) = 15(–3)(a3 • a2) = –45a(3 + 2) = –45a5 7. 2c2 + 15c + 7 = (2c + 1)(c + 7)Check: (2c + 1)(c + 7) = 2c2 + 14c + 1c + 7= 2c2 + 15c + 7 8. 15t2– 26t + 11 = (15t– 11)(t– 1)Check: (15t– 11)(t– 1) = 15t2– 15t– 11t + 11= 15t2– 26t + 11 9. 2q2 + 11q + 5 =(2q + 1)(q + 5)Check: (2q + 1)(q + 5) = 2q2 + 10q + 1q + 5= 2q2 + 11q + 5 12-4

  11. a. • Multiply the numerators and multiply the denominators. 8 y2 8 y2 56 y3 • = b. • 7 y 7 y x – 2 x – 6 x – 2 x – 6 x x + 5 x x + 5 Multiply the numerators and multiply the denominators. Leave the answer in factored form. x(x – 2) (x + 5) (x – 6) • = Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 Multiply. 12-4

  12. 3x + 1 4 3x + 1 4 • = • Factor denominator. 2 1 8x (3x – 1) (3x + 1) 8x (3x – 1) (3x + 1) 3x +1 4 • = Divide out the common factors (3x +1) and 4. 1 1 8x 9x2 – 1 8x 9x2 – 1 2x 3x – 1 = Simplify. Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 3x +1 4 Multiply and . 12-4

  13. 5x + 1 3x + 12 5x + 1 3x + 12 5x + 1 3 (x + 4) 5x + 1 3 (x + 4) (x + 3) (x + 4) 1 Factor. • (x2 + 7x + 12) = • 1 Divide out the common factor x + 4. • = 1 (x + 3) (x + 4) 1 (5x +1) (x + 3) 3 = Leave in factored form. Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 Multiply and x2 + 7x + 12. 12-4

  14. x2 + 13x +40 x – 7 x2 + 13x +40 x – 7 x2 + 13x +40 x – 7 Multiply by , the reciprocal of . x2 – 49 x + 8 ÷ = • x + 8 x2 – 49 (x + 5) (x + 8) x – 7 (x + 7) (x – 7) x + 8 x + 8 x2 – 49 x2 – 49 x + 8 x + 8 x2 – 49 = • Factor. 1 Divide out the common factors x + 8 and x – 7. 1 (x + 8) (x + 5) x – 7 (x + 7) (x – 7) x + 8 = • 1 1 = (x + 5) (x + 7) Leave in factored form. Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 Divide by . 12-4

  15. Multiply by the reciprocal of 8x2 + 16x x2 + 9x + 14 11x 8x2 + 16x 1 x2 + 9x + 14 11x 1 8x2 + 16x 1 8x (x + 2) 1 8x (x + 2) = ÷ • (x + 7) (x + 2) 11x Factor. = • 1 Divide out the common factor x + 2. (x + 7) (x + 2) 11x • = 1 x + 7 88x2 Simplify. = Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 x2 + 9x + 14 11x Divide by (8x2 + 16x). 12-4

  16. • x2 + 9x + 18 2x + 1 6x + 3 x + 6 • 7x2 5 15 14x x + 3 x + 1 9x2 x + 2 4x + 8 3x (x2 + 5x + 6) ÷ • 1 (x + 1)(x + 2) 2x + 4 x2 + 11x + 18 x + 1 x2 + 14x + 45 ÷ x + 9 x2 + 20x + 99 (x2 + 12x + 11) • Multiplying and Dividing Rational Expressions ALGEBRA 1 LESSON 12-4 Multiply or divide. 1. 2. 3x 2 3(x + 3) 3. 4. 12x 5. 2(x + 5) x + 1 6. x + 1 12-4

  17. Dividing Polynomials ALGEBRA 1 LESSON 12-5 (For help, go to Lessons 9-1 and 9-3.) Write each polynomial in standard form. 1. 9a– 4a2 + 1 2. 3x2– 6 + 5x–x3 3.–2 + 8t Find each product. 4. (2x + 4)(x + 3) 5. (–3n – 4)(n– 5) 6. (3a2 + 1)(2a– 7) 12-5

  18. Dividing Polynomials ALGEBRA 1 LESSON 12-5 Solutions 1. 9a– 4a2 + 1 = –4a2 + 9a + 1 2. 3x2– 6 + 5x – x3 = –x3 + 3x2 + 5x– 6 3.–2 + 8t = 8t– 2 4. (2x + 4)(x + 3) = (2x)(x) + (2x)(3) + (4)(x) + (4)(3)= 2x2 + 6x + 4x + 12 = 2x2 + 10x + 12 5. (–3n– 4)(n– 5) = (–3n)(n) + (–3n)(–5) + (–4)(n) + (–4)(–5)= –3n2 + 15n– 4n + 20 = –3n2 + 11n + 20 6. (3a2 + 1)(2a– 7) = (3a2)(2a) + (3a2)(–7) + (1)(2a) + (1)(–7)= 6a3– 21a2 + 2a– 7 12-5

  19. Multiply by the reciprocal of 3x2. 1 3x2 (18x3 + 9x2 – 15x) ÷ 3x2 = (18x3 + 9x2 – 15x) • . 18x3 3x2 9x2 3x2 15x 3x2 Use the Distributive Property. = + – 5 x Use the division rules for exponents. = 6x1 + 3x0 – 5 x = 6x + 3 – Simplify. Dividing Polynomials ALGEBRA 1 LESSON 12-5 Divide (18x3 + 9x2 – 15x) by 3x2. 12-5

  20. Step 1: Begin the long division process. Align terms by their degree. So put 5x above 2x of the dividend. 5x x + 2 5x2 + 2x – 3 Divide: Think 5x2 ÷ x = 5x. – 8x – 3 Bring down – 3. 5x2 + 10x Multiply: 5x(x + 2) = 5x2 + 10x. Then subtract. Dividing Polynomials ALGEBRA 1 LESSON 12-5 Divide (5x2 + 2x – 3) by (x + 2) 12-5

  21. 5x – 8 x + 2 5x2 + 2x – 3 5x2 + 10x – 8x – 3 Divide: –8x ÷ x = – 8 – 8x – 16 Multiply: – 8(x + 2) = – 8x – 16. Then subtract. 13 The remainder is 13. The answer is 5x – 8 + . 13 x + 2 Dividing Polynomials ALGEBRA 1 LESSON 12-5 (continued) Step 2: Repeat the process: Divide, multiply, subtract, bring down. 12-5

  22. Since A = w, divide the area by the width to find the length. 3x2 + 2x + 3 2x – 3 6x3 – 5x2 + 0x – 9 Rewrite the dividend with 0x. 6x3 – 9x2 4x2 + 0x 4x2 + 6x –6x – 9 –6x – 9 0 Dividing Polynomials ALGEBRA 1 LESSON 12-5 The width and area of a rectangle are shown in the figure below. What is the length? The length of the rectangle is (3x2 + 2x + 3) in. 12-5

  23. – 2 6x 6x2 – 6x 4 x – 1 x – 1 6x2 – 8x – 2 –2x –2 –2x + 2 –4 The answer is 6x – 2 – . Dividing Polynomials ALGEBRA 1 LESSON 12-5 Divide (–8x – 2 + 6x2) by (–1 + x). Rewrite –8x – 2 + 6x2 as 6x2 – 8x – 2 and –1 + x as x – 1. Then divide. 12-5

  24. 3x2 – 2x + 3 + 16x2 – 12x + 9 + 2 4x + 3 2 2x + 3 Dividing Polynomials ALGEBRA 1 LESSON 12-5 Divide. 1. (x8 – x6 + x4) ÷ x22. (4x2 – 2x – 6) ÷ (x + 1) 3. (6x3 + 5x2 + 11) ÷ (2x + 3) 4. (29 + 64x3) ÷ (4x + 3) x6 – x4 + x2 4x – 6 12-5

  25. 2. 3 7 5 7 – 1. 4 9 2 9 + 4. 5 12 6. 3 4 – 5 6 2 9 5. 1 4 1 3 + – 9. 7. 8. 7x 12 x 12 4x 9 2x 9 – – + 7 12y 1 12y 10.x2 + 3x + 2 11.y2 + 7y + 12 12.t2 – 4t + 4 Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 (For help, go to Lessons 1-5 and 906.) Simplify each expression. 3. 1 2 5 2 + – Factor each quadratic expression. 12-6

  26. 1. 4 + 2 9 6 9 3 • 2 3 • 3 2 3 = = = = 2 7 2. 3 7 5 7 3 – 5 7 –2 7 – = = = – 4 9 2 9 + 4. 5 6 2 9 5. 1 4 1 3 + 3. – 1 18 5 • 3 6 • 3 2 • 2 9 • 2 15 18 19 18 1 2 = + = or 1 = + 4 18 4 12 + 5 2 – 3 – 4 12 –1 12 1 • 3 4 • 3 1 • 4 3 • 4 3 12 1 12 1 + (– 5) 7 –4 2 = – = – = = = – = = = –2 Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 Solutions 6. 5 12 3 4 5 12 3 • 3 4 • 3 5 12 9 12 5 – 9 12 –4 12 1 3 – = – = – = = = – 12-6

  27. 4x + 2x 9 6x 9 2x 3 2 3 3 • 2 • x 3 • 3 1 2 x = = = = or x 7x – x 12 6x 12 x 2 6 • x 6 • 2 = = = = or 9. 7. 8. 7x 12 x 12 4x 9 2x 9 7 – 1 12y 6 12y 6 • 1 6 • 2y 1 2y – – + = = = = 1 12y 7 12y Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 Solutions (continued) 10. Factors of 2 with a sum of 3: 1 and 2.x2 + 3x + 2 = (x + 1)(x + 2) 11. Factors of 12 with a sum of 7: 3 and 4.y2 + 7y + 12 = (y + 3)(y + 4) 12. Factors of 4 with a sum of –4: –2 and –2.t2 – 4t + 4 = (t – 2)(t – 2) or (t – 2)2 12-6

  28. + = Add the numerators. = Simplify the numerator. 4 + 2 x + 3 4 x + 3 2 x + 3 4 x + 3 2 x + 3 6 x + 3 Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 Add and . 12-6

  29. Subtract the numerators. = – 4x + 7 – (3x + 5) 3x2 + 2x – 8 4x + 7 – 3x + 5 3x2 + 2x – 8 Use the Distributive Property. = 3x + 5 3x2 + 2x – 8 3x + 5 3x2 + 2x – 8 4x + 7 3x2 + 2x – 8 4x + 7 3x2 + 2x – 8 = 1 Simplify the numerator. x + 2 3x2 + 2x – 8 1 3x – 4 Factor the denominator. Divide out the common factor x + 2. = x + 2 (3x – 4) (x + 2) 1 Simplify. = Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 Subtract from . 12-6

  30. 1 8 Step 1: Find the LCD of and . 4x = 2 • 2 • xFactor each denominator. 8 = 2 • 2 • 2 LCD = 2 • 2 • 2 • x = 8x 1 • x 8 • x = + 2 • 3 2 • 4x 3 4x 3 4x 3 4x Rewrite each fraction using the LCD. 1 8 + = + Simplify numerators and denominators. 6 8x x 8x 6 + x 8x = Add the numerators. Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 1 8 Add + . Step 2: Rewrite using the LCD and add. 12-6

  31. Rewrite the fractions using the LCD. 3 (x + 4) (x + 4) (x – 5) + = + 3 x – 5 3 x – 5 7 (x – 5) (x + 4) (x – 5) 7 x + 4 7 x + 4 Simplify each numerator. = + 3x + 12 (x + 4) (x – 5) 7x – 35 (x + 4) (x – 5) 7x – 35 + 3x + 12 (x + 4) (x – 5) Add the numerators. = 10x – 23 (x + 4) (x – 5) = Simplify the numerator. Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 Add and . Step 1:  Find the LCD of x + 4 and x – 5. Since there are no common factors, the LCD is (x + 4)(x – 5). Step 2: Rewrite using the LCD and add. 12-6

  32. 5415 r distance rate Miami to Seattle time: time = 14% more than a number is 114% of the number. 5415 1.14r distance rate Seattle to Miami time: time = Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 The distance between Seattle, Washington, and Miami, Florida, is about 5415 miles. The ground speed for jet traffic from Seattle to Miami can be about 14% faster than the ground speed from Miami to Seattle. Use r for the jet’s ground speed. Write and simplify an expression for the round-trip air time. 12-6

  33. 6173 1.14r 5415 1.14r 5415 r 5415 1.14r + + Rewrite using the LCD, 1.14r. Add the numerators. 11588 1.14r = 10165 r = Simplify. Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 (continued) 5415 r 5415 1.14r An expression for the total time is + 12-6

  34. 1 9 2. Add + . 1. Add + . 2 5x 6y – 7 y + 2 2y – 3 y + 2 3. Subtract – . 4. Subtract – . 8 x2 – 4 3 x2 – 4 x – 4 x + 3 5. Add 8 + . 5 x2 – 4 2 3x + 4 6 x + 2 4(5x + 7) (3x + 4)(x + 2) Adding and Subtracting Rational Expressions ALGEBRA 1 LESSON 12-6 18 + 5x 45x 4(y – 1) y + 2 9x + 20 x + 3 12-6

  35. 1. 3. m 3 27 m 2. = 1 x 3 5 3 t 5 2 = = 1 y2 1 8y 5 6 4. 5. 6. ; ; 4 3x 2 5 3 4n 1 3x 1 2 2 n ; ; ; ; Solving Rational Equations ALGEBRA 1 LESSON 12-7 (For help, go to Lessons 4-1 and 12-6.) Solve each proportion. Find the LCD of each group of expressions. 12-7

  36. 1 x 3 5 = m 3 27 m = 3 t 5 2 5t = 6 3x = 5 = 5 3 2 3 6 5 1 5 x = or 1 t = or 1 Solving Rational Equations ALGEBRA 1 LESSON 12-7 Solutions 1. 2. 3. m2 = 81 m = ± 81 = ± 9 4. 4n = 2 • 2 • n; 2 = 2; n = n; LCD = 2 • 2 • n = 4n 5. 3x = 3 • x; 5 = 5; 3x = 3 • x; LCD = 3 • 5 • x = 15x 6. 8y = 2 • 2 • 2 • y; y2 = y • y; 6 = 2 • 3; LCD = 2 • 2 • 2 • 3 • y • y = 24y2 12-7

  37. The denominators are 8, x, and 2x. The LCD is 8x. 3 8 4 x 14 2x + = Multiply each side by the LCD. 8x = 8x 3 8 4 x 4 + 1 14 2x 14 2x 1 3 8 4 x No rational expressions. Now you can solve. 8x = 8x 8x + Use the Distributive Property. 3x + 32 = 56 1 1 1 3x = 24 Subtract 32 from each side. Divide each side by 3, then simplify. x = 8 Solving Rational Equations ALGEBRA 1 LESSON 12-7 3 8 4 x 14 2x Solve + = . 12-7

  38. 3 8 4 8 14 2(8) Check: + 7 8 14 16 7 8 7 8 = Solving Rational Equations ALGEBRA 1 LESSON 12-7 (continued) 12-7

  39. 5 x – 1 x2 Multiply each side by the LCD, x2. x2 = Use the Distributive Property. x 1 5 x 6 = 5x – x2 Simplify. x2 – x2 (1) x2 = 1 1 x2 – 5x + 6 = 0 Collect like terms on one side. 6 x2 6 x2 6 x2 Solving Rational Equations ALGEBRA 1 LESSON 12-7 5 x Solve = – 1. Check the solution. (x – 3)(x – 2) = 0 Factor the quadratic expression. x – 3 = 0 or x – 2 = 0 Use the Zero-Product Property. x = 3 or x = 2 Solve. 12-7

  40. 5 2 5 3 – 1 – 1 3 2 3 2 = 2 3 2 3 = 6 32 6 22 Solving Rational Equations ALGEBRA 1 LESSON 12-7 (continued) Check: 12-7

  41. Define: Let n = the time to complete the job if they work together (in minutes). Person Work Rate Time Worked Part of (part of job/min.) (min) Job Done Renee n Joanne n 1 20 1 30 n 20 n 30 Relate: Renee’s part done + Joanne’s part done = complete job. Solving Rational Equations ALGEBRA 1 LESSON 12-7 Renee can mow the lawn in 20 minutes. Joanne can do the same job in 30 minutes. How long will it take them if they work together? 12-7

  42. n 20 n 30 + Write: = 1 60 + Multiply each side by the LCD, 60. = 60(1) n 20 n 30 5n = 60 Simplify. n = 12 Simplify. Check:  Renee will do • = of the job, and Joanne will do of the job. Together, they will do = 1, or the whole job. 1 20 12 1 3 5 12 1 2 5 = • 1 30 3 5 2 5 + Solving Rational Equations ALGEBRA 1 LESSON 12-7 (continued) Relate: Renee’s part done + Joanne’s part done = complete job. 3n + 2n = 60 Use the Distributive Property. It will take two of them 12 minutes to mow the lawn working together. 12-7

  43. 4(x + 8) = x2(1) Write cross products. 1 x + 8 1 x + 8 = 4 x2 4 x2 x – 8 = 0  or  x + 4 = 0 Use the Zero-Product Property. x = 8  or x = –4 Solve. Solving Rational Equations ALGEBRA 1 LESSON 12-7 Solve = . Check the solution. 4x + 32 = x2Use the Distributive Property. x2 – 4x – 32 = 0 Collect terms on one side. (x – 8)(x + 4) = 0 Factor the quadratic expression. 12-7

  44. 1 –4 + 8 1 8 + 8 1 x + 8 = 4 82 4 x2 Check: 4 (–4)2 1 16 1 4 4 16 4 64 = = Solving Rational Equations ALGEBRA 1 LESSON 12-7 (continued) 12-7

  45. = (x + 3) (x – 1) = 4(x – 1) Write the cross products. 4 x – 1 4 x – 1 x + 3 x – 1 x + 3 x – 1 x – 1 = 0 Use the Zero-Product Property. x = 1 Simplify. Solving Rational Equations ALGEBRA 1 LESSON 12-7 Solve = . x2 – x + 3x – 3 = 4x – 4 Use the Distributive Property. x2 + 2x – 3 = 4x – 4 Combine like terms. x2 – 2x + 1 = 0 Subtract 4x – 4 from each side. (x – 1) (x – 1) = 0 Factor. 12-7

  46. 4 1 – 1 Undefined! There is no division by 0. 1 + 3 1 – 1 Check: 4 0 4 0 = Solving Rational Equations ALGEBRA 1 LESSON 12-7 (continued) The equation has no solution because 1 makes a denominator equal 0. 12-7

  47. 2 5 1. Solve = . 2. Solve = . 3. Solve = . 4. Solve + = . 5. Juanita can wash the car in 30 minutes. Gabe can wash the car in 40 minutes. Working together, how long will it take? 1 x 2 x + 1 x 4x + 3 1 3 –2 3(2 + x) 1 2x 1 7 4 x + 3 1 2 17 min Solving Rational Equations ALGEBRA 1 LESSON 12-7 –2 1 –4 1, 3 12-7

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