Section 8.1 Solutions

1. No, you can either have zero, one, or infinitely many. Examine graphs.

3. This means there is no realistic break-even point. By the time the company produces one unit they are already making profit.

5. You can solve by substitution (isolating [latex]x[/latex] or [latex]y[/latex] ), graphically, or by addition.

7. Yes

9. Yes

11. [latex]\left(-1,2\right)[/latex]

13. [latex]\left(-3,1\right)[/latex]

15. [latex]\left(-\frac{3}{5},0\right)[/latex]

17. No solutions exist.

19. [latex]\left(\frac{72}{5},\frac{132}{5}\right)[/latex]

21. [latex]\left(6,-6\right)[/latex]

23. [latex]\left(-\frac{1}{2},\frac{1}{10}\right)[/latex]

25. No solutions exist.

27. [latex]\left(-\frac{1}{5},\frac{2}{3}\right)[/latex]

29. [latex]\left(x,\frac{x+3}{2}\right)[/latex]

31. [latex]\left(-4,4\right)[/latex]

33. [latex]\left(\frac{1}{2},\frac{1}{8}\right)[/latex]

35. [latex]\left(\frac{1}{6},0\right)[/latex]

37. [latex]\left(x,2\left(7x - 6\right)\right)[/latex]

39. [latex]\left(-\frac{5}{6},\frac{4}{3}\right)[/latex]

41. Consistent with one solution

43. Consistent with one solution

45. Dependent with infinitely many solutions

47. [latex]\left(-3.08,4.91\right)[/latex]

49. [latex]\left(-1.52,2.29\right)[/latex]

51. [latex]\left(\frac{A+B}{2},\frac{A-B}{2}\right)[/latex]

53. [latex]\left(\frac{-1}{A-B},\frac{A}{A-B}\right)[/latex]

55. [latex]\left(\frac{CE-BF}{BD-AE},\frac{AF-CD}{BD-AE}\right)[/latex]

57. They never turn a profit.

59. [latex]\left(1,250,100,000\right)[/latex]

61. The numbers are 7.5 and 20.5.

63. 24,000

65. 790 sophomores, 805 freshman

67. 56 men, 74 women

69. 10 gallons of 10% solution, 15 gallons of 60% solution

71. Swan Peak: $750,000, Riverside: $350,000

73. $12,500 in the first account, $10,500 in the second account.

75. High-tops: 45, Low-tops: 15

77. Infinitely many solutions. We need more information.

Section 8.2 Solutions

1. A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

3. No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

5. Choose any number between each solution and plug into [latex]C\left(x\right)[/latex] and [latex]R\left(x\right)[/latex]. If [latex]C\left(x\right)<R\left(x\right),\text{}[/latex] then there is profit.

7. [latex]\left(0,-3\right),\left(3,0\right)[/latex]

9. [latex]\left(-\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right),\left(\frac{3\sqrt{2}}{2},-\frac{3\sqrt{2}}{2}\right)[/latex]

11. [latex]\left(-3,0\right),\left(3,0\right)[/latex]

13. [latex]\left(\frac{1}{4},-\frac{\sqrt{62}}{8}\right),\left(\frac{1}{4},\frac{\sqrt{62}}{8}\right)[/latex]

15. [latex]\left(-\frac{\sqrt{398}}{4},\frac{199}{4}\right),\left(\frac{\sqrt{398}}{4},\frac{199}{4}\right)[/latex]

17. [latex]\left(0,2\right),\left(1,3\right)[/latex]

19. [latex]\left(-\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right)[/latex]

21. [latex]\left(5,0\right)[/latex]

23. [latex]\left(0,0\right)[/latex]

25. [latex]\left(3,0\right)[/latex]

27. No Solutions Exist

29. No Solutions Exist

31. [latex]\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right),\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right),\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right),\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)[/latex]

33. [latex]\left(2,0\right)[/latex]

35. [latex]\left(-\sqrt{7},-3\right),\left(-\sqrt{7},3\right),\left(\sqrt{7},-3\right),\left(\sqrt{7},3\right)[/latex]

37. [latex]\left(-\sqrt{\frac{1}{2}\left(\sqrt{73}-5\right)},\frac{1}{2}\left(7-\sqrt{73}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{73}-5\right)},\frac{1}{2}\left(7-\sqrt{73}\right)\right)[/latex]

39.

41.

43.

45.

47.

49. [latex]\left(-2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(-2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right)[/latex]

51. No Solution Exists

53. [latex]x=0,y>0[/latex] and [latex]0<x<1,\sqrt{x}<y<\frac{1}{x}[/latex]

55. 12, 288

57. 2–20 computers

Section 8.3 Solutions

1. No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, [latex]\frac{1}{{x}^{2}+1}[/latex] cannot be decomposed because the denominator cannot be factored.

3. Graph both sides and ensure they are equal.

5. If we choose [latex]x=-1[/latex], then the B-term disappears, letting us immediately know that [latex]A=3[/latex]. We could alternatively plug in [latex]x=-\frac{5}{3}[/latex], giving us a B-value of [latex]-2[/latex].

7. [latex]\frac{8}{x+3}-\frac{5}{x - 8}[/latex]

9. [latex]\frac{1}{x+5}+\frac{9}{x+2}[/latex]

11. [latex]\frac{3}{5x - 2}+\frac{4}{4x - 1}[/latex]

13. [latex]\frac{5}{2\left(x+3\right)}+\frac{5}{2\left(x - 3\right)}[/latex]

15. [latex]\frac{3}{x+2}+\frac{3}{x - 2}[/latex]

17. [latex]\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x - 3\right)}[/latex]

19. [latex]\frac{8}{x - 3}-\frac{5}{x - 2}[/latex]

21. [latex]\frac{1}{x - 2}+\frac{2}{{\left(x - 2\right)}^{2}}[/latex]

23. [latex]-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}[/latex]

25. [latex]-\frac{1}{x - 7}-\frac{2}{{\left(x - 7\right)}^{2}}[/latex]

27. [latex]\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}[/latex]

29. [latex]\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}[/latex]

31. [latex]\frac{x+1}{{x}^{2}+x+3}+\frac{3}{x+2}[/latex]

33. [latex]\frac{4 - 3x}{{x}^{2}+3x+8}+\frac{1}{x - 1}[/latex]

35. [latex]\frac{2x - 1}{{x}^{2}+6x+1}+\frac{2}{x+3}[/latex]

37. [latex]\frac{1}{{x}^{2}+x+1}+\frac{4}{x - 1}[/latex]

39. [latex]\frac{2}{{x}^{2}-3x+9}+\frac{3}{x+3}[/latex]

41. [latex]-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x - 3}[/latex]

43. [latex]\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}[/latex]

45. [latex]\frac{x+6}{{x}^{2}+1}+\frac{4x+3}{{\left({x}^{2}+1\right)}^{2}}[/latex]

47. [latex]\frac{x+1}{x+2}+\frac{2x+3}{{\left(x+2\right)}^{2}}[/latex]

49. [latex]\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}[/latex]

51. [latex]\frac{1}{8x}-\frac{x}{8\left({x}^{2}+4\right)}+\frac{10-x}{2{\left({x}^{2}+4\right)}^{2}}[/latex]

53. [latex]-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x - 1}-\frac{7}{{\left(x - 1\right)}^{2}}[/latex]

55. [latex]\frac{1}{x+1}-\frac{2}{{\left(x+1\right)}^{2}}+\frac{5}{{\left(x+1\right)}^{3}}[/latex]

57. [latex]\frac{5}{x - 2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x - 8\right)}[/latex]

59. [latex]-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}[/latex]