Bài tập cơ bản giải phương trình vô tỷ – Toán lớp 9

Bài tập cơ bản phương trình vô tỷ dành cho các em học sinh lớp 9 tự giải ôn luyện chuyên đề thường xuất hiện trong các đề thi vào 10 này.

Bài 1: Giải các phương trình sau:
1. \sqrt{{{x}^{2}}-6x+9}=x

2. \sqrt{4{{x}^{2}}-12x+9}=x-1

3. x+\sqrt{4{{x}^{2}}-4x+1}=2

4. 1-\sqrt{4{{x}^{4}}-20{{x}^{2}}+25}=0

5. {{x}^{2}}-\sqrt{{{x}^{2}}}=0

6. {{x}^{2}}+\sqrt{{{x}^{2}}}=0

7. x+\sqrt{{{x}^{2}}-4x+4}=0

8. x-\sqrt{4{{x}^{2}}-12x+9}=0

9. 3x-1-\sqrt{4{{x}^{2}}-12x+9}=0

10. x-\sqrt{4{{x}^{2}}-12x+9}=3

11. \sqrt{3-2\sqrt{2}}-\sqrt{{{x}^{2}}+2x\sqrt{3}+3}=0

12. \sqrt{5{{x}^{2}}-2x\sqrt{5}+1}=\sqrt{6-2\sqrt{5}}

13.\sqrt{4{{x}^{2}}+4x\sqrt{7}+7}-\sqrt{8-2\sqrt{7}}=0

14. \sqrt{7-2\sqrt{10}}-\sqrt{5{{x}^{2}}-2x\sqrt{10}+2}=0

15. \sqrt{11+6\sqrt{2}}=\sqrt{2{{x}^{2}}-6x\sqrt{2}+9}

16. \sqrt{11-\sqrt{120}}=\sqrt{5{{x}^{2}}+x\sqrt{120}+6}

17. \sqrt{1+2x\sqrt{3}+3{{x}^{2}}}-\sqrt{3+2x\sqrt{3}+{{x}^{2}}}=0

18. \sqrt{5{{x}^{2}}-2x\sqrt{5}+1}-\sqrt{4{{x}^{2}}+4x\sqrt{5}+5}=0

19. \sqrt{16{{x}^{2}}+8x\sqrt{2}+2}-\sqrt{9{{x}^{2}}-6x\sqrt{2}+2}=0

20. \sqrt{2{{x}^{2}}-2x\sqrt{6}+3}-\sqrt{2-2x\sqrt{6}+3{{x}^{2}}}=0

21. \sqrt{8{{x}^{2}}-4x\sqrt{2}+1}-\sqrt{{{x}^{2}}-6x\sqrt{2}+18}=0

22. \sqrt{5{{x}^{2}}+2x\sqrt{30}+6}-\sqrt{6{{x}^{2}}+2x\sqrt{30}+5}=0

23. \sqrt{{{x}^{2}}}=x

24. \sqrt{{{x}^{2}}-2x+1}=x-1

25.\sqrt{{{x}^{2}}-4x+4}=x-2

26. \sqrt{16-8x+{{x}^{2}}}=4-x

27. \sqrt{4{{x}^{2}}-12x+9}=2x-3

28. \sqrt{25{{x}^{2}}-10x+x}=5x-1

29. \sqrt{{{x}^{2}}-2x\sqrt{5}+5}=x-\sqrt{5}

30. \sqrt{3{{x}^{2}}-6x\sqrt{2}+6}=\sqrt{3}x-\sqrt{6}

31. \sqrt{10{{x}^{2}}-12x\sqrt{10}+36}=\sqrt{10}x-6

32. \sqrt{7{{x}^{2}}+2x\sqrt{14}+2}=\sqrt{7}x+\sqrt{2}

33. \sqrt{{{x}^{2}}}=-x

34. \sqrt{{{x}^{2}}-6x+9}=3-x

35. \sqrt{{{x}^{2}}-4x+4}=2-x

36. \sqrt{{{x}^{2}}+4x+4}=-x-2

37. \sqrt{4{{x}^{2}}+4x+1}=-2x-1

38. \sqrt{{{x}^{2}}+x+\frac{1}{4}}=-x-\frac{1}{2}

39. \sqrt{x+2\sqrt{x}+1}-\sqrt{x-2\sqrt{x}+1}=2

40. \sqrt{x+4\sqrt{x}+4}+\sqrt{x-4\sqrt{x}+4}=4

41. \sqrt{x+6\sqrt{x}+9}-6=\sqrt{9-6\sqrt{x}+x}

42. \sqrt{4x+4\sqrt{x}+1}=\sqrt{1-4\sqrt{x}+4x}+2

43. \sqrt{x-2\sqrt{x-1}}-\sqrt{x+2\sqrt{x-1}}=-2

44. \sqrt{x-2\sqrt{x-2}-1}-\sqrt{x+2+4\sqrt{x-2}+3}=0

45. -\sqrt{x+3+4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=-5

46. \sqrt{x+4\sqrt{x-4}}-\sqrt{x-4\sqrt{x-4}}=4

47. -\sqrt{2x+4+6\sqrt{2x-5}}+\sqrt{2x-4-2\sqrt{2x-5}}=-4

48. 4+\sqrt{2x+6-6\sqrt{2x-3}}=\sqrt{2x-2+2\sqrt{2x-3}}

49. \sqrt{x+2\sqrt{x}+1}+\sqrt{x-2\sqrt{x}+1}=2

50. \sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}=2

51. \sqrt{x-2\sqrt{x-2}-1}+\sqrt{x+2+4\sqrt{x-2}}-3=0

52. \sqrt{x+3+4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=5

53. \sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}=4

54. \sqrt{2x+4+6\sqrt{2x-5}}+\sqrt{2x-4-2\sqrt{2x-5}}=4

55. \sqrt{2x+2\sqrt{2x}+1}-\sqrt{2x-2\sqrt{2x}+1}=2

56. \sqrt{4+4\sqrt{3x}+3x}-4=\sqrt{4-4\sqrt{3x}+3x}

57. \sqrt{12x-4\sqrt{3}x+1}-\sqrt{1+4\sqrt{3}x+12x}=-2

58. \sqrt{18x+6\sqrt{2}x+1}-\sqrt{1-6\sqrt{2}x+18x}=2.

Bài 2: Giải các phương trình sau:

1. \sqrt{2x-3}=\sqrt{x-1}

2.\sqrt{2x-3}-\sqrt{x+3}=0

3. \sqrt{x-1}=\sqrt{2x+3}

4. \sqrt{2x-3}=\sqrt{x+1}

5. \sqrt{x+2}=\sqrt{2x-4}

6. \sqrt{2-x}=\sqrt{3+x}

7. \sqrt{1-x}-\sqrt{x-3}=0

8. \sqrt{-2x}-\sqrt{2-x}=0

9. \sqrt{6-x}-\sqrt{-3x}=0

10. \sqrt{{{x}^{2}}-3x}-\sqrt{5\left( 3-x \right)}=0

11. \sqrt{x-2}=\sqrt{x-2}

12. \displaystyle \sqrt{4x-8}=2\sqrt{x-2}

13. \sqrt{x-1}-\sqrt{x-4}=0

14. \sqrt{2-x}-\sqrt{3-x}=0

15. \sqrt{{{x}^{2}}-5}=\sqrt{4x-9}

16. \sqrt{2{{x}^{2}}-6x+2}=\sqrt{{{x}^{2}}-3x}

17. \sqrt{{{x}^{2}}-2x-4}=\sqrt{2-x}

18. \sqrt{{{x}^{2}}-x-1}=\sqrt{x-1}

19. \sqrt{x-2}-\sqrt{{{x}^{2}}-2x}=0

20. \sqrt{{{x}^{2}}-x-2}=\sqrt{x+1}

21. \sqrt{2{{x}^{2}}-10x+11}=\sqrt{{{x}^{2}}-6x+8}

22. \sqrt{2{{x}^{2}}+6x-3}=\sqrt{{{x}^{2}}+4x}

23. \sqrt{2{{x}^{2}}+x-9}=\sqrt{{{x}^{2}}-x-6}

24. \sqrt{x-1}=2

25. \sqrt{2x-3}=13

26. \sqrt{2x-3}=\sqrt{2}

27. \sqrt{x\left( x-2 \right)}-\sqrt{3}=0

28. 3\sqrt{x}=-1

29. \sqrt{x-1}+2=0

30. 6-\sqrt{2x+3}=12

31. 3\sqrt{{{x}^{2}}-x}-\sqrt{54}=0

32. 2\sqrt{2}-\sqrt{{{x}^{2}}+2x}=0

33. 2\sqrt{3}-\sqrt{7x-{{x}^{2}}}=0

34. 3-\sqrt{{{x}^{2}}+3}=0

35. 2-\sqrt{x\left( 4-x \right)}=0

36. 3-\sqrt{-x\left( x+6 \right)}=0

37. 2-\sqrt{{{x}^{2}}-1}=0

38. 1-\sqrt{{{x}^{2}}-2}=0

39. \sqrt{16}-2\sqrt{{{x}^{2}}+3x}=0

40. 2\sqrt{3}-\sqrt{x\left( x+7 \right)}=0

41. \sqrt{3-x}=3x-5( PTNK, CD, 1999-2000)

42. x-\sqrt{4x-3}=2 ( PTNK, AB, 2004-2005, Vòng 1)

43. \sqrt{{{x}^{2}}-x}=x

44. \sqrt{{{x}^{2}}-1}=x-1

45. \sqrt{3-{{x}^{2}}}=x

46. \sqrt{{{x}^{2}}-2x+2}=x-1

47. \sqrt{5-{{x}^{2}}}=x-1( LÊ HỒNG PHONG, 2006-2007, Vòng 1)

48. x-2=\sqrt{{{x}^{2}}-4x+3}

49. \sqrt{x}-x=0

50. \sqrt{x}+x=0

51. x-\sqrt{2x-9}=6

52. 2x-\sqrt{4x-1}=0

53. 3x-\sqrt{6x-1}=0

54. x-2\sqrt{x-1}=16

55. x+\sqrt{-\left( 2x+1 \right)}=0

56. x+2\sqrt{x-1}=0

57. x+\sqrt{3\left( 6-x \right)}=0

58. x+\sqrt{x+3}=0

59. x+\sqrt{2x+3}=0

60. x+\sqrt{5-4x}=0

61. 2x+\sqrt{3x+7}=0

62. 3x+\sqrt{5x+4}=0

63. 2x-\sqrt{x\left( 1-2x \right)}=1

64. x+\sqrt{1-{{x}^{2}}}=1

65. x+\sqrt{4-{{x}^{2}}}=2

66. 2x+\sqrt{4\left| x \right|-1}=0

67. x+2\sqrt{\left| x \right|-1}=0

68. -5x+\sqrt{2\left| x \right|+3}=0

69. 3\sqrt{-2\left| x \right|+1}=9x

70. 7x+\sqrt{\left| x \right|-4}=0

71. \sqrt{10\left| x \right|-10}-6x=0

72. \sqrt{-3\left| x \right|+2}+1=x

73. \sqrt{7\left| x \right|+11}+x+1=0

74. \sqrt{2\left| x-1 \right|-3}-x+1=0

75. -3\sqrt{2\left| 2x+1 \right|-5}+6x+3=0

76. 5\sqrt{2\left| 1-5x \right|+3}-5+25x=0

77. -2\sqrt{-8\left| 2-3x \right|+9}+4-6x=0

78. 7\sqrt{-4\left| 3x-9 \right|+5}+21x-63=0

79. \sqrt{-2\left| 1-2x \right|+3}+1=2x

80. \displaystyle 3\sqrt{2x}-5\sqrt{8x}+7\sqrt{18x}=8

81. \frac{5}{3}\sqrt{15x}-\sqrt{15x}+11=\frac{1}{3}\sqrt{15x}

82. \sqrt{4x+20}-\sqrt{x+5}-\frac{1}{3}\sqrt{9x+45}=4

83. \sqrt{36x-36}-\sqrt{9x-9}-\sqrt{4x-4}=16-\sqrt{x-1}

84. \sqrt{36x-72}-\sqrt{9x-18}+\sqrt{4x-8}+\sqrt{x-2}=\sqrt{72}

85. \sqrt{9x+18}-\sqrt{x+2}-\sqrt{4x+8}+2\sqrt{x+5}=0

86. \frac{5}{3}\sqrt{45x}-\sqrt{125x}-\frac{1}{3}\sqrt{405x}-2\sqrt{16-16x}=0

87. \frac{1}{5}\sqrt{25x-125}-\frac{3}{2}\sqrt{x-5}+\sqrt{36x-180}+\sqrt{9x-27}=0

88. \sqrt{36x-216}-\sqrt{x-6}+\frac{7}{2}\sqrt{4x-24}=\frac{3}{7}\sqrt{49x-343}

89. 15\sqrt{x-7}-2\sqrt{9x-63}-9\sqrt{25x-175}=\sqrt{4x-24}

90. \displaystyle \sqrt{49x-98}-\sqrt{9x-18}-\sqrt{16x-32}=\sqrt{4x-4}

91. 7\sqrt{x}+\sqrt{81x-81}+\sqrt{x-1}=\sqrt{100x-100}

92. \sqrt{{{x}^{2}}-2x\sqrt{5}+5}+\sqrt{{{x}^{2}}-x\sqrt{5}}=0

93. \sqrt{0,2{{x}^{2}}-2x+5}+\sqrt{3{{x}^{2}}-15x}=0

94. \sqrt{{{x}^{2}}+4x}+\sqrt{\frac{{{x}^{2}}}{2}-8}=0

95. \sqrt{x-1}+\sqrt{{{x}^{2}}-3x+2}=0

96. \sqrt{2+x}+\sqrt{4{{x}^{2}}-6x-10}=0

97. \sqrt{{{x}^{2}}-9}+\sqrt{{{x}^{2}}-4x+3}=0

98. \sqrt{-2{{x}^{2}}+3x+5}+\sqrt{2{{x}^{2}}-7x-15}=0

LUYỆN TẬP
Bài 3: Giải các phương trình sau:

1. {{x}^{2}}+4\left( \left| x-2 \right|-x \right)-1=0
a) Đặt t=\left| x-2 \right| để đưa phương trình trên về phương trình theo ẩn t.
b) Tìm t rồi sau đó tìm x.

2. {{x}^{2}}+2\left( \left| x-1 \right|-x \right)-2=0

3. {{x}^{2}}+3\left( \left| x-3 \right|-2x \right)-9=0

4. {{x}^{2}}+4\left( \left| x-4 \right|-2x \right)+4=0

5. {{x}^{2}}+\left( x+\left| x+1 \right| \right)-14=0 (PTNK BAN CD 2006-2007)

6. 7-4\sqrt{2x-1}=3\left| 1-2x \right|
a) Đặt t=\sqrt{3-2x} để đưa phương trình trên về phương trình theo ẩn t.
b) Tìm t rồi sau đó tìm x.

7. 5-\sqrt{3-2x}=\left| 2x-3 \right| (PTNK BAN CD 2000-2001)

8. 20-\sqrt{3-2x}=\left| 2x-3 \right| (PTNK BAN CD 2004-2005)

9. 12-\sqrt{4-3x}=\left| 3x-4 \right| (PTNK BAN CD 2007-2008)

10. \left| 1-2x \right|=6-\sqrt{2x-1}

11. 2\left| 4-3x \right|+3\sqrt{3x-4}-2=0

12. 3\left| 3x-1 \right|+8\sqrt{1-3x}=3

13. 2\sqrt{\frac{6x-1}{2x}}=\frac{2x}{6x-1}+1
a) Đặt t=\sqrt{\frac{6x-1}{2x}} để đưa phương trình trên về phương trình theo ẩn t.
b) Tìm t rồi sau đó tìm x.

14. 2\sqrt{\frac{3x-1}{x}}=\frac{x}{3x-1}+1 (PTNK BAN CD 2001-2002)

15. 2\sqrt{\frac{6x-1}{2x}}=\frac{2x}{6x-1}+1

16. 2\sqrt{\frac{9x-1}{3x}}=\frac{3x}{9x-1}+1

17. 2\sqrt{\frac{3x-3}{x}}=\frac{3x}{x-1}+1

18. \sqrt{\frac{6x-4}{x}}=\frac{x}{3x-2}+1

19. \sqrt{\frac{2x-1}{x}}+1+\sqrt{\frac{x}{2x-1}}=3\frac{x}{2x-1}

20. 2\sqrt{\frac{x}{x-1}}-\sqrt{\frac{x-1}{x}}=2\frac{x-1}{x}+3

21. 3\sqrt{\frac{2x}{x-1}}+4\sqrt{\frac{x-1}{2x}}=3\frac{x-1}{2x}+10

22. \sqrt{\frac{x}{3-2x}}+5\sqrt{\frac{3-2x}{x}}=4\frac{3-2x}{x}+5

23. 2\sqrt{\frac{2x+1}{x}}-3\sqrt{\frac{x}{2x+1}}=4\frac{x}{2x+1}+7

24. \frac{2+\sqrt{19-2x}}{x}=1

25. \sqrt{x}-\frac{4}{\sqrt{2+x}}+\sqrt{2+x}=0

26. \sqrt{9-5x}=\sqrt{3-x}+\frac{6}{\sqrt{3-x}}

27. \sqrt{2-x}+\frac{4}{3+\sqrt{2-x}}=2

28. \frac{\left( 5-x \right)\sqrt{5-x}+\left( x-3 \right)\sqrt{x-3}}{\sqrt{5-x}+\sqrt{x-3}}=2
Bài 4: Giải các phương trình sau:

1. \sqrt{2x+3}+\sqrt{2x+2}=1

2. \sqrt{1-x}+\sqrt{4+x}=3

3. \sqrt{x+4}-\sqrt{2x-6}=1 ( PTNK, AB, 2006-2007, Vòng 1)

4. \sqrt{3x-5}+\sqrt{2x+3}=\sqrt{x+2} (HSG, Q. Tân Bình, 2003-2004)

5. \sqrt{x-2}+\sqrt{x-1}=\sqrt{2x-3} ( LÊ HỒNG PHONG 2001-2002, Vòng 1)

6. \sqrt{3x+7}-\sqrt{x+1}=2

7. \sqrt{x+3}+\sqrt{x-1}=2

8. \sqrt{x+5}+\sqrt{3-x}=4

9. \sqrt{{{x}^{2}}+9}-\sqrt{{{x}^{2}}+7}=2

10. \sqrt{2-{{x}^{2}}}+\sqrt{{{x}^{2}}+8}=4

11. \sqrt{x+3}-\sqrt{7-x}=\sqrt{2x-8}

12. \sqrt{2-x}=\sqrt{7-x}-\sqrt{-3-2x}

13. \sqrt{11-x}-\sqrt{x-1}=2

14. \sqrt{{{x}^{2}}+3x+2}-\sqrt{{{x}^{2}}+x+1}=1

15. \sqrt{5x-1}=\sqrt{3x-2}-\sqrt{2x-3}

16. \sqrt{5x-1}-\sqrt{x-1}=\sqrt{2x-4}

17. \sqrt{x+2}-\sqrt{2x-3}=\sqrt{3x-5} (HSG, Q.I,1999-2000)

18. \sqrt{x+4}-\sqrt{1-x}=\sqrt{1-2x}

19. \sqrt{x+9}=5-\sqrt{2x+4}

20. \sqrt{3x+4}-\sqrt{2x+1}=\sqrt{x+3}

21. \sqrt{x+12}+\sqrt{x-6}-\sqrt{x+2}-\sqrt{x-4}=0

22. \sqrt{3x+6}+\sqrt{3x-3}-\sqrt{3x+1}-\sqrt{3x-2}=0

23. \sqrt{x+4}+\sqrt{x-5}-\sqrt{x-1}-\sqrt{x-4}=0

24. \sqrt{2x+6}+\sqrt{2x-3}-\sqrt{2x+1}-\sqrt{2x-2}=0

25. \sqrt{x+6}+\sqrt{x-3}-\sqrt{x+1}-\sqrt{x-2}=0 (PTNK, AB, 2005-2006, Vòng 1)

26. \left\{ \begin{array}{l}x-y=5\\\sqrt{2x+1}-\sqrt{y+2}=2\end{array} \right. ( PTNK, AB, 2005-2006, Vòng 1)

27. \sqrt{x+\sqrt{x+11}}+\sqrt{x-\sqrt{x+11}}=4

28. \sqrt{x-\sqrt{x-2}}+\sqrt{x+\sqrt{x-2}}=3

29. \frac{2-\sqrt{x}}{2-x}=\sqrt{\frac{2}{2-x}}

30. \sqrt{\frac{20+x}{x}}+\sqrt{\frac{20-x}{x}}=\sqrt{6} ( đặt t=\frac{20}{x})

Bài 5: Giải các phương trình sau:

1. \sqrt{2x+1}+\sqrt{x-3}=2\sqrt{x}

2. \sqrt{x+2}-\sqrt{2x-3}=\sqrt{4x-7}

3. \sqrt{x}+\sqrt{x-3}=\sqrt{3\left( x-1 \right)}

4. \sqrt{x-2}+\sqrt{4-x}=\sqrt{6-x}

5. \sqrt{x+5}=2\sqrt{x}-\sqrt{2x-7}

6. \sqrt{3x+12}-\sqrt{4x+13}=\sqrt{x+1}

7. \sqrt{9-x}-\sqrt{x+4}=\sqrt{3x+1}

8. \sqrt{3x+4}=2\sqrt{x}-\sqrt{x-4}

9. \sqrt{2x+5}=\sqrt{12x+25}-\sqrt{5x+6}

10. \sqrt{x+1}+\sqrt{x-1}=\sqrt{3x-1}

11. \sqrt{3x-5}+\sqrt{2x-3}=\sqrt{x+2}

12. \sqrt{{{x}^{2}}+9}-\sqrt{{{x}^{2}}-7}=2

13. \sqrt{{{x}^{2}}+5}+\sqrt{{{x}^{2}}-3}=4

14. \sqrt{{{x}^{2}}-3x+6}+\sqrt{{{x}^{2}}-3x+3}=3

15. \sqrt{3{{x}^{2}}-2x+15}+\sqrt{3{{x}^{2}}-2x+8}=7

16. \sqrt{3{{x}^{2}}+5x+8}-\sqrt{3{{x}^{2}}+5x+1}=1

Bài 6: Giải các phương trình sau:

1. \sqrt{x+6}+\sqrt{x-3}-\sqrt{x+1}-\sqrt{x-2}=0

2. \sqrt{{{x}^{2}}+x-5}+\sqrt{{{x}^{2}}+8x-4}=5

3. \sqrt{3{{x}^{2}}+6x+16}+\sqrt{{{x}^{2}}+2x}=2\sqrt{{{x}^{2}}+2x+4}

4. \sqrt{2{{x}^{2}}-9x+4}+3\sqrt{2x-1}=\sqrt{2{{x}^{2}}+21x-11}

Bài 7: Giải các bất phương trình sau:

1. \sqrt{3x-5{{x}^{2}}}\le 5x-2 (PTNK, AB, 2006-2007, Vòng 1)

2. \sqrt{5-x}\le 2x-7

3. \sqrt{x+1}<2x-1 (PTNK, AB, 2002-2003, Vòng 1)

4. \sqrt{{{x}^{2}}-x-12}<7-x

5. 1-x+\sqrt{2{{x}^{2}}-3x-5}<0

6. \sqrt{{{x}^{2}}-3x-10}\le x-2

7. 3\sqrt{-{{x}^{2}}+x+6}+2\left( 2x-1 \right)<0

8. \sqrt{3{{x}^{2}}+13x+4}+2-x\le 0

9. 2\sqrt{3x+{{x}^{2}}}\le 2x-1

10. \sqrt{{{x}^{2}}-3x+3}\le 2x-1

11. \sqrt{{{x}^{2}}+3x-3}\le 2x-3

12. -6x+7\ge \sqrt{{{x}^{2}}-6x+7}

13. \sqrt{{{x}^{2}}-3x-10}\le 8-x

14. \sqrt{{{x}^{2}}+5x-6}<2x+3

15. \sqrt{x+3}<x+1

16. \sqrt{2x+12}<x+2

17. \displaystyle \sqrt{{{x}^{2}}-6x-40}\le 16-x

18. \sqrt{{{x}^{2}}+10x+4}\le 2x-4

19. 3\sqrt{-{{x}^{2}}+3x+54}+4x-6<0

20. 2-x+\sqrt{2{{x}^{2}}-6x-20}<0

Bài 8: Giải các bất phương trình sau:

1. \sqrt{3x-5{{x}^{2}}}\ge 5x-2

2. \sqrt{5-x}\ge 2x-7

3. \sqrt{x+1}\ge 2x-1

4. \sqrt{{{x}^{2}}-x-12}\ge 7-x

5. 1-x+\sqrt{2{{x}^{2}}-3x-5}>0

6. \sqrt{{{x}^{2}}-3x-10}\ge x-2

7. 3\sqrt{-{{x}^{2}}+x+6}+2\left( 2x-1 \right)>0

8. \sqrt{3{{x}^{2}}+13x+4}+2-x\ge 0

9. 2\sqrt{3x+{{x}^{2}}}\ge 2x-1

10. \sqrt{{{x}^{2}}-3x+3}\ge 2x-1

11. \sqrt{{{x}^{2}}+3x-3}\le 2x-3

12. -4x+2\le 4\sqrt{{{x}^{2}}-6x+5}

13. \sqrt{{{x}^{2}}-3x-10}\ge 8-x

14. \sqrt{{{x}^{2}}+5x-6}>2x+3

15. \sqrt{x+3}>x+1

16. \sqrt{2x+12}>x+2

17. \displaystyle \sqrt{{{x}^{2}}-6x-40}\ge 16-x

18. \sqrt{{{x}^{2}}+10x+4}\ge 2x-4

19. 3\sqrt{-{{x}^{2}}+3x+54}+4x-6>0

20. 2-x+\sqrt{2{{x}^{2}}-6x-20}>0

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