Chapter 8 Solutions to Odd-Numbered Problems

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)

41.
A shaded figure with a dotted line that has two marked points. The first point is at square root of two minus 1, two times (the square root of two minus one). The second point is at negative one minus square root of two, negative two times (one plus the square root of two).

43.
Two dotted, shaded figures with points marked. The first point is (negative square root of 37 over 2, 3 times square root of seven over two). The second point is (square root of 37 over 2, 3 times square root of 7 over two). The third point is (negative square root 37 over 2, negative 3 times square root 7 divided by 2). The fourth point is (square root 37 over 2, negative 3 times square root of 7 over two).

45.
Two dotted, shaded figures with marked points. The first point is negative square root of nineteen-tenths, square root of forty-seven-tenths. The second point is square root of 19 tenths, square root of 47 tenths. The third point is negative square root of 19 tenths, negative square root of 47 tenths. The fourth point is square root of 19 tenths, negative square root of 47 tenths.

47.
Two solid curving lines. The region below the blue line and to the right of the y axis is shaded.

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]0Section 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]